diff --git a/PackageInfo.g b/PackageInfo.g index 238288f5..c355191a 100644 --- a/PackageInfo.g +++ b/PackageInfo.g @@ -8,8 +8,8 @@ SetPackageInfo( rec( PackageName := "HAP", Subtitle := "Homological Algebra Programming", - Version := "1.61", - Date := "02/01/2024", + Version := "1.62", + Date := "01/02/2024", License := "GPL-2.0-or-later", SourceRepository := rec( diff --git a/README.md b/README.md index 6bde485c..d5799fad 100644 --- a/README.md +++ b/README.md @@ -32,12 +32,12 @@ Please send your bug reports to graham.ellis(at)nuigalway.ie . On a Linux machine with GAP (and optionally Polymake) installed, the HAP library can be loaded as follows: -* First download the file hap1.61.tar.gz to the subdirectory "pkg/" of GAP. (If +* First download the file hap1.62.tar.gz to the subdirectory "pkg/" of GAP. (If you don't have access to this, then create a directory "pkg" in your home directory and download the file there.) -* Change to directory "pkg/" and type "gunzip hap1.61.tar.gz" followed by -"tar -xvf hap1.61.tar" . +* Change to directory "pkg/" and type "gunzip hap1.62.tar.gz" followed by +"tar -xvf hap1.62.tar" . * Start GAP. (If you have created "pkg" in your home directory then start GAP with the command "gap -l 'path/homedir;' " where path/homedir is the path to @@ -46,12 +46,12 @@ your home directory.) * In GAP type " LoadPackage("HAP"); " . * Help on HAP can be found on the HAP home page (a version of which is -included in directory "pkg/Hap1.61/www" of this distribution). +included in directory "pkg/Hap1.62/www" of this distribution). * Performance can be significantly improved by using a compiled version of the HAP library. A compiled version can be created by the following steps. -1. Change to the directory "pkg/Hap1.61/" . +1. Change to the directory "pkg/Hap1.62/" . 2. Edit the file "compile" so that: PKGDIR is equal to the path to the directory "pkg" where your GAP packages are stored; GACDIR is equal to the path to the directory where the GAP compiler "gac" is stored. @@ -60,4 +60,4 @@ path to the directory where the GAP compiler "gac" is stored. The next time HAP is loaded a compiled version will be loaded. * Should you want to return to an uncompiled version, change to the directory -"pkg/Hap1.61/" and type "./uncompile". +"pkg/Hap1.62/" and type "./uncompile". diff --git a/_data/package.yml b/_data/package.yml index 5999130d..770ac2ed 100644 --- a/_data/package.yml +++ b/_data/package.yml @@ -1,7 +1,7 @@ name: HAP -version: "1.61" +version: "1.62" license: "GPL-2.0-or-later" -date: 2024-01-02 +date: 2024-02-01 description: | Homological Algebra Programming @@ -66,7 +66,7 @@ packageinfo: https://gap-packages.github.io/hap/PackageInfo.g downloads: - name: .tar.gz - url: https://github.com/gap-packages/hap/releases/download/v1.61/hap-1.61.tar.gz + url: https://github.com/gap-packages/hap/releases/download/v1.62/hap-1.62.tar.gz abstract: | This package provides some functions for group cohomology and algebraic topology. @@ -80,7 +80,7 @@ citeas: |

[Ell24] Ellis, G., HAP, Homological Algebra Programming, - Version 1.61 + Version 1.62 (2024)
(Refereed GAP package), https://gap-packages.github.io/hap. @@ -88,10 +88,10 @@ citeas: | bibtex: | - @misc{ HAP1.61, + @misc{ HAP1.62, author = {Ellis, G.}, - title = {{HAP}, Homological Algebra Programming, {V}ersion 1.61}, - month = {Jan}, + title = {{HAP}, Homological Algebra Programming, {V}ersion 1.62}, + month = {Feb}, year = {2024}, note = {Refereed GAP package}, howpublished = {\href {https://gap-packages.github.io/hap} diff --git a/doc/chap0.html b/doc/chap0.html index a0ca971c..1b47ccb9 100644 --- a/doc/chap0.html +++ b/doc/chap0.html @@ -28,9 +28,9 @@

Homological Algebra Programming

HAP

-

Version 1.61

+

Version 1.62

-

02 Jan 2024 +

01 Feb 2024

@@ -44,7 +44,7 @@

HAP

School of Mathematics
National University of Irelnd, Galway
Galway
(Ireland)

Abstract

-

HAP is a homological algebra library for use with the GAP computer algebra system, and is still under development. The current version 1.61 was released on 02 Jan 2024 .
The initial focus of the library was on computations related to the cohomology of finite and infinite groups, with particular emphasis on integral coefficients. The focus has since broadened to include Steenrod algebras of finite groups, Bredon homology, cohomology of simplicial groups, and general computations in algebraic topology relating to finite CW-complexes, covering spaces, knots, knotted surfaces, and topics such as persitent homology arising in topological data analysis.
This document describes the functions available in HAP. Examples illustrating these functions are available in the HAP tutorial.

+

HAP is a homological algebra library for use with the GAP computer algebra system, and is still under development. The current version 1.62 was released on 01 Feb 2024 .
The initial focus of the library was on computations related to the cohomology of finite and infinite groups, with particular emphasis on integral coefficients. The focus has since broadened to include Steenrod algebras of finite groups, Bredon homology, cohomology of simplicial groups, and general computations in algebraic topology relating to finite CW-complexes, covering spaces, knots, knotted surfaces, and topics such as persitent homology arising in topological data analysis.
This document describes the functions available in HAP. Examples illustrating these functions are available in the HAP tutorial.

Copyright

@@ -836,14 +836,16 @@

Contents


  34.2-2 IdentityArrow
  34.2-3 InitialArrow
  34.2-4 TerminalArrow -
  34.2-5 Source -
  34.2-6 Target -
  34.2-7 CategoryName -
  34.2-8 CompositionEqualityAdditionMinus -
  34.2-9 Object -
  34.2-10 Mapping -
  34.2-11 IsCategoryObject -
  34.2-12 IsCategoryArrow +
  34.2-5 HasInitialObject +
  34.2-6 HasTerminalObject +
  34.2-7 Source +
  34.2-8 Target +
  34.2-9 CategoryName +
  34.2-10 CompositionEqualityAdditionMinus +
  34.2-11 Object +
  34.2-12 Mapping +
  34.2-13 IsCategoryObject +
  34.2-14 IsCategoryArrow
35 Arrays and Pseudo lists diff --git a/doc/chap0.txt b/doc/chap0.txt index c526e5de..0fd1ba8d 100644 --- a/doc/chap0.txt +++ b/doc/chap0.txt @@ -6,10 +6,10 @@ HAP - Version 1.61 + Version 1.62 - 02 Jan 2024 + 01 Feb 2024 Graham Ellis @@ -29,8 +29,8 @@ ------------------------------------------------------- Abstract HAP is a homological algebra library for use with the GAP computer algebra - system, and is still under development. The current version 1.61 was - released on 02 Jan 2024 . + system, and is still under development. The current version 1.62 was + released on 01 Feb 2024 . The initial focus of the library was on computations related to the cohomology of finite and infinite groups, with particular emphasis on integral coefficients. The focus has since broadened to include Steenrod @@ -654,14 +654,16 @@ 34.2-2 IdentityArrow 34.2-3 InitialArrow 34.2-4 TerminalArrow - 34.2-5 Source - 34.2-6 Target - 34.2-7 CategoryName - 34.2-8 CompositionEqualityAdditionMinus - 34.2-9 Object - 34.2-10 Mapping - 34.2-11 IsCategoryObject - 34.2-12 IsCategoryArrow + 34.2-5 HasInitialObject + 34.2-6 HasTerminalObject + 34.2-7 Source + 34.2-8 Target + 34.2-9 CategoryName + 34.2-10 CompositionEqualityAdditionMinus + 34.2-11 Object + 34.2-12 Mapping + 34.2-13 IsCategoryObject + 34.2-14 IsCategoryArrow 35 Arrays and Pseudo lists 35.1   35.1-1 Array diff --git a/doc/chap0_mj.html b/doc/chap0_mj.html index 16fb18be..fb3b4de0 100644 --- a/doc/chap0_mj.html +++ b/doc/chap0_mj.html @@ -31,9 +31,9 @@

Homological Algebra Programming

HAP

-

Version 1.61

+

Version 1.62

-

02 Jan 2024 +

01 Feb 2024

@@ -47,7 +47,7 @@

HAP

School of Mathematics
National University of Irelnd, Galway
Galway
(Ireland)

Abstract

-

HAP is a homological algebra library for use with the GAP computer algebra system, and is still under development. The current version 1.61 was released on 02 Jan 2024 .
The initial focus of the library was on computations related to the cohomology of finite and infinite groups, with particular emphasis on integral coefficients. The focus has since broadened to include Steenrod algebras of finite groups, Bredon homology, cohomology of simplicial groups, and general computations in algebraic topology relating to finite CW-complexes, covering spaces, knots, knotted surfaces, and topics such as persitent homology arising in topological data analysis.
This document describes the functions available in HAP. Examples illustrating these functions are available in the HAP tutorial.

+

HAP is a homological algebra library for use with the GAP computer algebra system, and is still under development. The current version 1.62 was released on 01 Feb 2024 .
The initial focus of the library was on computations related to the cohomology of finite and infinite groups, with particular emphasis on integral coefficients. The focus has since broadened to include Steenrod algebras of finite groups, Bredon homology, cohomology of simplicial groups, and general computations in algebraic topology relating to finite CW-complexes, covering spaces, knots, knotted surfaces, and topics such as persitent homology arising in topological data analysis.
This document describes the functions available in HAP. Examples illustrating these functions are available in the HAP tutorial.

Copyright

@@ -839,14 +839,16 @@

Contents


  34.2-2 IdentityArrow
  34.2-3 InitialArrow
  34.2-4 TerminalArrow -
  34.2-5 Source -
  34.2-6 Target -
  34.2-7 CategoryName -
  34.2-8 CompositionEqualityAdditionMinus -
  34.2-9 Object -
  34.2-10 Mapping -
  34.2-11 IsCategoryObject -
  34.2-12 IsCategoryArrow +
  34.2-5 HasInitialObject +
  34.2-6 HasTerminalObject +
  34.2-7 Source +
  34.2-8 Target +
  34.2-9 CategoryName +
  34.2-10 CompositionEqualityAdditionMinus +
  34.2-11 Object +
  34.2-12 Mapping +
  34.2-13 IsCategoryObject +
  34.2-14 IsCategoryArrow
35 Arrays and Pseudo lists diff --git a/doc/chap1.html b/doc/chap1.html index 3afa6b21..d233749d 100644 --- a/doc/chap1.html +++ b/doc/chap1.html @@ -226,7 +226,7 @@
1.1-6 PurePermutahedralComplex
‣ PurePermutahedralComplex( A )( function )

Inputs a binary array A and returns the pure permutahedral complex represented by A.

-

Examples: 1 , 2 , 3 

+

Examples: 1 , 2 , 3 , 4 

@@ -323,7 +323,7 @@
1.1-16 ReadImageAsPureCubicalComplex
‣ ReadImageAsPureCubicalComplex( str, t )( function )

Reads an image file identified by a string str such as "file.bmp", "file.eps", "file.jpg", "path/file.png" etc., together with an integer t between 0 and 765. It returns a 2-dimensional pure cubical complex corresponding to a black/white version of the image determined by the threshold t. The 2-cells of the pure cubical complex correspond to pixels with RGB value R+G+B ≤ t.

-

Examples: 1 , 2 , 3 , 4 

+

Examples: 1 , 2 , 3 , 4 , 5 

@@ -683,7 +683,7 @@
1.4-7 PureComplexBoundary
‣ PureComplexBoundary( M )( function )

Inputs a d-dimensional pure cubical or pure permutahedral complex M and returns a d-dimensional complex consisting of the closure of those d-cells whose boundaries contains some cell with coboundary of size less than the maximal possible size.

-

Examples:

+

Examples: 1 

@@ -703,7 +703,7 @@
1.4-9 PureComplexDifference
‣ PureComplexDifference( M, N )( function )

Inputs two pure cubical complexes or two pure permutahedral complexes and returns the difference M - N.

-

Examples:

+

Examples: 1 

@@ -733,7 +733,7 @@
1.4-12 PureComplexUnion
‣ PureComplexUnion( M, N )( function )

Inputs two pure cubical complexes or two pure permutahedral complexes and returns their union.

-

Examples:

+

Examples: 1 

@@ -843,7 +843,7 @@
1.5-5 FundamentalGroup

Inputs a regular CW map F and returns the induced homomorphism of fundamental groups. If the number of some zero cell in the domain of F is entered as an optional second variable then the fundamental group is based at this zero cell.

-

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 

+

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 

@@ -1162,7 +1162,7 @@
1.10-4 Display
‣ Display( M )( function )

Displays a graph G; a 2- or 3-dimensional pure cubical complex M; a 3-dimensional pure permutahedral complex M.

-

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 

+

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 

diff --git a/doc/chap1.txt b/doc/chap1.txt index bf8dd28c..e79376bd 100644 --- a/doc/chap1.txt +++ b/doc/chap1.txt @@ -101,8 +101,8 @@ Inputs a binary array A and returns the pure permutahedral complex represented by A. - Examples: 1 (../tutorial/chap2.html) , 2 - (../www/SideLinks/About/aboutPeripheral.html) , 3 + Examples: 1 (../tutorial/chap2.html) , 2 (../tutorial/chap5.html) , 3 + (../www/SideLinks/About/aboutPeripheral.html) , 4 (../www/SideLinks/About/aboutCubical.html)  1.1-7 CayleyGraphOfGroup @@ -224,9 +224,9 @@ threshold t. The 2-cells of the pure cubical complex correspond to pixels with RGB value R+G+B ≤ t. - Examples: 1 (../tutorial/chap10.html) , 2 - (../www/SideLinks/About/aboutPersistent.html) , 3 - (../www/SideLinks/About/aboutCubical.html) , 4 + Examples: 1 (../tutorial/chap5.html) , 2 (../tutorial/chap10.html) , 3 + (../www/SideLinks/About/aboutPersistent.html) , 4 + (../www/SideLinks/About/aboutCubical.html) , 5 (../www/SideLinks/About/aboutTDA.html)  1.1-17 ReadImageAsFilteredPureCubicalComplex @@ -700,7 +700,7 @@ whose boundaries contains some cell with coboundary of size less than the maximal possible size. - Examples: + Examples: 1 (../tutorial/chap5.html)  1.4-8 PureComplexComplement @@ -724,7 +724,7 @@ Inputs two pure cubical complexes or two pure permutahedral complexes and returns the difference M - N. - Examples: + Examples: 1 (../tutorial/chap5.html)  1.4-10 PureComplexInterstection @@ -754,7 +754,7 @@ Inputs two pure cubical complexes or two pure permutahedral complexes and returns their union. - Examples: + Examples: 1 (../tutorial/chap5.html)  1.4-13 SimplifiedComplex @@ -883,12 +883,13 @@ Examples: 1 (../tutorial/chap1.html) , 2 (../tutorial/chap2.html) , 3 (../tutorial/chap3.html) , 4 (../tutorial/chap4.html) , 5 - (../tutorial/chap5.html) , 6 (../www/SideLinks/About/aboutLinks.html) , 7 - (../www/SideLinks/About/aboutPeripheral.html) , 8 - (../www/SideLinks/About/aboutCoveringSpaces.html) , 9 - (../www/SideLinks/About/aboutCoverinSpaces.html) , 10 - (../www/SideLinks/About/aboutQuandles.html) , 11 - (../www/SideLinks/About/aboutRandomComplexes.html) , 12 + (../tutorial/chap5.html) , 6 (../tutorial/chap11.html) , 7 + (../www/SideLinks/About/aboutLinks.html) , 8 + (../www/SideLinks/About/aboutPeripheral.html) , 9 + (../www/SideLinks/About/aboutCoveringSpaces.html) , 10 + (../www/SideLinks/About/aboutCoverinSpaces.html) , 11 + (../www/SideLinks/About/aboutQuandles.html) , 12 + (../www/SideLinks/About/aboutRandomComplexes.html) , 13 (../www/SideLinks/About/aboutKnots.html)  1.5-6 FundamentalGroupOfQuotient @@ -1377,20 +1378,20 @@ Examples: 1 (../tutorial/chap1.html) , 2 (../tutorial/chap2.html) , 3 (../tutorial/chap4.html) , 4 (../tutorial/chap5.html) , 5 - (../tutorial/chap7.html) , 6 (../tutorial/chap9.html) , 7 - (../tutorial/chap10.html) , 8 (../tutorial/chap11.html) , 9 - (../www/SideLinks/About/aboutMetrics.html) , 10 - (../www/SideLinks/About/aboutArtinGroups.html) , 11 - (../www/SideLinks/About/aboutNoncrossing.html) , 12 - (../www/SideLinks/About/aboutPeriodic.html) , 13 - (../www/SideLinks/About/aboutPersistent.html) , 14 - (../www/SideLinks/About/aboutPolytopes.html) , 15 - (../www/SideLinks/About/aboutQuandles2.html) , 16 - (../www/SideLinks/About/aboutQuandles.html) , 17 - (../www/SideLinks/About/aboutSuperperfect.html) , 18 - (../www/SideLinks/About/aboutGraphsOfGroups.html) , 19 - (../www/SideLinks/About/aboutIntro.html) , 20 - (../www/SideLinks/About/aboutKnotsQuandles.html) , 21 + (../tutorial/chap6.html) , 6 (../tutorial/chap7.html) , 7 + (../tutorial/chap9.html) , 8 (../tutorial/chap10.html) , 9 + (../tutorial/chap11.html) , 10 (../www/SideLinks/About/aboutMetrics.html) , + 11 (../www/SideLinks/About/aboutArtinGroups.html) , 12 + (../www/SideLinks/About/aboutNoncrossing.html) , 13 + (../www/SideLinks/About/aboutPeriodic.html) , 14 + (../www/SideLinks/About/aboutPersistent.html) , 15 + (../www/SideLinks/About/aboutPolytopes.html) , 16 + (../www/SideLinks/About/aboutQuandles2.html) , 17 + (../www/SideLinks/About/aboutQuandles.html) , 18 + (../www/SideLinks/About/aboutSuperperfect.html) , 19 + (../www/SideLinks/About/aboutGraphsOfGroups.html) , 20 + (../www/SideLinks/About/aboutIntro.html) , 21 + (../www/SideLinks/About/aboutKnotsQuandles.html) , 22 (../www/SideLinks/About/aboutTopology.html)  1.10-5 DisplayArcPresentation diff --git a/doc/chap17.html b/doc/chap17.html index 9de30de4..86c7e010 100644 --- a/doc/chap17.html +++ b/doc/chap17.html @@ -63,7 +63,7 @@
17.1-2 IdentityAmongRelatorsDisplay

This function uses GraphViz software.

-

Examples: 1 , 2 

+

Examples: 1 , 2 , 3 

@@ -101,7 +101,7 @@
17.1-4 PresentationOfResolution

where G is isomorphic to F modulo the normal closure of S. This presentation for G corresponds to the 2-skeleton of the classifying CW-space from which R was constructed. The resolution R requires no contracting homotopy.

-

Examples: 1 , 2 , 3 

+

Examples: 1 , 2 , 3 , 4 

diff --git a/doc/chap17.txt b/doc/chap17.txt index 901bab76..69fa66ae 100644 --- a/doc/chap17.txt +++ b/doc/chap17.txt @@ -37,7 +37,8 @@ This function uses GraphViz software. - Examples: 1 (../www/SideLinks/About/aboutPeriodic.html) , 2 + Examples: 1 (../tutorial/chap6.html) , 2 + (../www/SideLinks/About/aboutPeriodic.html) , 3 (../www/SideLinks/About/aboutTopology.html)  17.1-3 IsAspherical @@ -80,8 +81,9 @@ for G corresponds to the 2-skeleton of the classifying CW-space from which R was constructed. The resolution R requires no contracting homotopy. - Examples: 1 (../www/SideLinks/About/aboutPolytopes.html) , 2 - (../www/SideLinks/About/aboutSpaceGroup.html) , 3 + Examples: 1 (../tutorial/chap6.html) , 2 + (../www/SideLinks/About/aboutPolytopes.html) , 3 + (../www/SideLinks/About/aboutSpaceGroup.html) , 4 (../www/SideLinks/About/aboutTopology.html)  17.1-5 TorsionGeneratorsAbelianGroup diff --git a/doc/chap17_mj.html b/doc/chap17_mj.html index 1ce73592..77f35896 100644 --- a/doc/chap17_mj.html +++ b/doc/chap17_mj.html @@ -66,7 +66,7 @@
17.1-2 IdentityAmongRelatorsDisplay

This function uses GraphViz software.

-

Examples: 1 , 2 

+

Examples: 1 , 2 , 3 

@@ -104,7 +104,7 @@
17.1-4 PresentationOfResolution

where \(G\) is isomorphic to \(F\) modulo the normal closure of \(S\). This presentation for \(G\) corresponds to the 2-skeleton of the classifying CW-space from which \(R\) was constructed. The resolution \(R\) requires no contracting homotopy.

-

Examples: 1 , 2 , 3 

+

Examples: 1 , 2 , 3 , 4 

diff --git a/doc/chap1_mj.html b/doc/chap1_mj.html index a6a022ef..701d21ba 100644 --- a/doc/chap1_mj.html +++ b/doc/chap1_mj.html @@ -229,7 +229,7 @@
1.1-6 PurePermutahedralComplex
‣ PurePermutahedralComplex( A )( function )

Inputs a binary array \(A\) and returns the pure permutahedral complex represented by \(A\).

-

Examples: 1 , 2 , 3 

+

Examples: 1 , 2 , 3 , 4 

@@ -326,7 +326,7 @@
1.1-16 ReadImageAsPureCubicalComplex
‣ ReadImageAsPureCubicalComplex( str, t )( function )

Reads an image file identified by a string str such as "file.bmp", "file.eps", "file.jpg", "path/file.png" etc., together with an integer \(t\) between \(0\) and \(765\). It returns a \(2\)-dimensional pure cubical complex corresponding to a black/white version of the image determined by the threshold \(t\). The \(2\)-cells of the pure cubical complex correspond to pixels with RGB value \(R+G+B \le t\).

-

Examples: 1 , 2 , 3 , 4 

+

Examples: 1 , 2 , 3 , 4 , 5 

@@ -686,7 +686,7 @@
1.4-7 PureComplexBoundary
‣ PureComplexBoundary( M )( function )

Inputs a \(d\)-dimensional pure cubical or pure permutahedral complex \(M\) and returns a \(d\)-dimensional complex consisting of the closure of those \(d\)-cells whose boundaries contains some cell with coboundary of size less than the maximal possible size.

-

Examples:

+

Examples: 1 

@@ -706,7 +706,7 @@
1.4-9 PureComplexDifference
‣ PureComplexDifference( M, N )( function )

Inputs two pure cubical complexes or two pure permutahedral complexes and returns the difference \( M - N\).

-

Examples:

+

Examples: 1 

@@ -736,7 +736,7 @@
1.4-12 PureComplexUnion
‣ PureComplexUnion( M, N )( function )

Inputs two pure cubical complexes or two pure permutahedral complexes and returns their union.

-

Examples:

+

Examples: 1 

@@ -846,7 +846,7 @@
1.5-5 FundamentalGroup

Inputs a regular CW map \(F\) and returns the induced homomorphism of fundamental groups. If the number of some zero cell in the domain of \(F\) is entered as an optional second variable then the fundamental group is based at this zero cell.

-

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 

+

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 

@@ -1165,7 +1165,7 @@
1.10-4 Display
‣ Display( M )( function )

Displays a graph \(G\); a \(2\)- or \(3\)-dimensional pure cubical complex \(M\); a \(3\)-dimensional pure permutahedral complex \(M\).

-

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 

+

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 

diff --git a/doc/chap2.html b/doc/chap2.html index c61245e3..ea8b7db4 100644 --- a/doc/chap2.html +++ b/doc/chap2.html @@ -152,7 +152,7 @@
2.1-6 ResolutionNilpotentGroup
‣ ResolutionNilpotentGroup( G, k )( function )

Inputs a nilpotent group G (which can be infinite) and an integer k ≥ 1. It returns k+1 terms of a free ZG-resolution of Z.

-

Examples: 1 , 2 , 3 , 4 

+

Examples: 1 , 2 , 3 , 4 , 5 

diff --git a/doc/chap2.txt b/doc/chap2.txt index b5c3b1a2..17798e30 100644 --- a/doc/chap2.txt +++ b/doc/chap2.txt @@ -107,9 +107,9 @@ Inputs a nilpotent group G (which can be infinite) and an integer k ≥ 1. It returns k+1 terms of a free ZG-resolution of Z. - Examples: 1 (../tutorial/chap11.html) , 2 - (../www/SideLinks/About/aboutCohomologyRings.html) , 3 - (../www/SideLinks/About/aboutRosenbergerMonster.html) , 4 + Examples: 1 (../tutorial/chap6.html) , 2 (../tutorial/chap11.html) , 3 + (../www/SideLinks/About/aboutCohomologyRings.html) , 4 + (../www/SideLinks/About/aboutRosenbergerMonster.html) , 5 (../www/SideLinks/About/aboutExtensions.html)  2.1-7 ResolutionNormalSeries diff --git a/doc/chap29.html b/doc/chap29.html index e4d2e85d..87bc1b47 100644 --- a/doc/chap29.html +++ b/doc/chap29.html @@ -154,7 +154,7 @@
29.1-8 ReadImageAsPureCubicalComplex
‣ ReadImageAsPureCubicalComplex( str, n )( function )

Reads an image file str (= "file.png", "file.eps", "file.bmp" etc) and an integer n between 0 and 765. It returns a 2-dimensional pure cubical complex based on the black/white version of the image determined by the threshold n.

-

Examples: 1 , 2 , 3 , 4 

+

Examples: 1 , 2 , 3 , 4 , 5 

@@ -407,7 +407,7 @@
29.1-35 HomotopyEquivalentMinimalPureCubicalSubcomplex
‣ HomotopyEquivalentMinimalPureCubicalSubcomplex( T, S )( function )

Inputs a pure cubical complex T together with a pure cubical subcomplex S. It returns a pure cubical subcomplex H of T which contains S and is minimal with respect to the property that it is homotopy equivalent to T.

-

Examples:

+

Examples: 1 

@@ -537,7 +537,7 @@
29.1-49 ComplementOfFilteredPureCubicalComplex
‣ ComplementOfFilteredPureCubicalComplex( M )( function )

Inputs a filtered pure cubical complex M and returns the complement as a filtered pure cubical complex.

-

Examples:

+

Examples: 1 

diff --git a/doc/chap29.txt b/doc/chap29.txt index 24dc9e29..310881e9 100644 --- a/doc/chap29.txt +++ b/doc/chap29.txt @@ -97,9 +97,9 @@ integer n between 0 and 765. It returns a 2-dimensional pure cubical complex based on the black/white version of the image determined by the threshold n. - Examples: 1 (../tutorial/chap10.html) , 2 - (../www/SideLinks/About/aboutPersistent.html) , 3 - (../www/SideLinks/About/aboutCubical.html) , 4 + Examples: 1 (../tutorial/chap5.html) , 2 (../tutorial/chap10.html) , 3 + (../www/SideLinks/About/aboutPersistent.html) , 4 + (../www/SideLinks/About/aboutCubical.html) , 5 (../www/SideLinks/About/aboutTDA.html)  29.1-9 ReadLinkImageAsPureCubicalComplex @@ -463,7 +463,7 @@ It returns a pure cubical subcomplex H of T which contains S and is minimal with respect to the property that it is homotopy equivalent to T. - Examples: + Examples: 1 (../tutorial/chap5.html)  29.1-36 BoundaryOfPureCubicalComplex @@ -618,7 +618,7 @@ Inputs a filtered pure cubical complex M and returns the complement as a filtered pure cubical complex. - Examples: + Examples: 1 (../tutorial/chap5.html)  29.1-50 PersistentHomologyOfFilteredPureCubicalComplex diff --git a/doc/chap29_mj.html b/doc/chap29_mj.html index e0c2b233..94c8a95a 100644 --- a/doc/chap29_mj.html +++ b/doc/chap29_mj.html @@ -157,7 +157,7 @@
29.1-8 ReadImageAsPureCubicalComplex
‣ ReadImageAsPureCubicalComplex( str, n )( function )

Reads an image file \(str\) (= "file.png", "file.eps", "file.bmp" etc) and an integer \(n\) between 0 and 765. It returns a 2-dimensional pure cubical complex based on the black/white version of the image determined by the threshold \(n\).

-

Examples: 1 , 2 , 3 , 4 

+

Examples: 1 , 2 , 3 , 4 , 5 

@@ -410,7 +410,7 @@
29.1-35 HomotopyEquivalentMinimalPureCubicalSubcomplex
‣ HomotopyEquivalentMinimalPureCubicalSubcomplex( T, S )( function )

Inputs a pure cubical complex \(T\) together with a pure cubical subcomplex \(S\). It returns a pure cubical subcomplex \(H\) of \(T\) which contains \(S\) and is minimal with respect to the property that it is homotopy equivalent to \(T\).

-

Examples:

+

Examples: 1 

@@ -540,7 +540,7 @@
29.1-49 ComplementOfFilteredPureCubicalComplex
‣ ComplementOfFilteredPureCubicalComplex( M )( function )

Inputs a filtered pure cubical complex \(M\) and returns the complement as a filtered pure cubical complex.

-

Examples:

+

Examples: 1 

diff --git a/doc/chap2_mj.html b/doc/chap2_mj.html index 368f4287..0354a5ea 100644 --- a/doc/chap2_mj.html +++ b/doc/chap2_mj.html @@ -155,7 +155,7 @@
2.1-6 ResolutionNilpotentGroup
‣ ResolutionNilpotentGroup( G, k )( function )

Inputs a nilpotent group \(G\) (which can be infinite) and an integer \(k \ge 1\). It returns \(k+1\) terms of a free \(\mathbb ZG\)-resolution of \(\mathbb Z\).

-

Examples: 1 , 2 , 3 , 4 

+

Examples: 1 , 2 , 3 , 4 , 5 

diff --git a/doc/chap30.html b/doc/chap30.html index ac14cc50..4ed145d3 100644 --- a/doc/chap30.html +++ b/doc/chap30.html @@ -99,7 +99,7 @@
30.1-6 FundamentalGroup
‣ FundamentalGroup( Y, n )( function )

Inputs a regular CW-complex Y and, optionally, the number of some 0-cell. It returns the fundamental group of Y based at the 0-cell n. The group is returned as a finitely presented group. If n is not specified then it is set n=1. The algorithm requires a discrete vector field on Y. If Y does not initially have a discrete vector field then one is constructed.

-

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 

+

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 

 [Top of Book]  [Contents]   [Previous Chapter]   [Next Chapter] 
diff --git a/doc/chap30.txt b/doc/chap30.txt index c14cca4e..eac30b12 100644 --- a/doc/chap30.txt +++ b/doc/chap30.txt @@ -88,11 +88,12 @@ Examples: 1 (../tutorial/chap1.html) , 2 (../tutorial/chap2.html) , 3 (../tutorial/chap3.html) , 4 (../tutorial/chap4.html) , 5 - (../tutorial/chap5.html) , 6 (../www/SideLinks/About/aboutLinks.html) , 7 - (../www/SideLinks/About/aboutPeripheral.html) , 8 - (../www/SideLinks/About/aboutCoveringSpaces.html) , 9 - (../www/SideLinks/About/aboutCoverinSpaces.html) , 10 - (../www/SideLinks/About/aboutQuandles.html) , 11 - (../www/SideLinks/About/aboutRandomComplexes.html) , 12 + (../tutorial/chap5.html) , 6 (../tutorial/chap11.html) , 7 + (../www/SideLinks/About/aboutLinks.html) , 8 + (../www/SideLinks/About/aboutPeripheral.html) , 9 + (../www/SideLinks/About/aboutCoveringSpaces.html) , 10 + (../www/SideLinks/About/aboutCoverinSpaces.html) , 11 + (../www/SideLinks/About/aboutQuandles.html) , 12 + (../www/SideLinks/About/aboutRandomComplexes.html) , 13 (../www/SideLinks/About/aboutKnots.html)  diff --git a/doc/chap30_mj.html b/doc/chap30_mj.html index 36e57254..29c31ec1 100644 --- a/doc/chap30_mj.html +++ b/doc/chap30_mj.html @@ -102,7 +102,7 @@
30.1-6 FundamentalGroup
‣ FundamentalGroup( Y, n )( function )

Inputs a regular CW-complex \(Y\) and, optionally, the number of some 0-cell. It returns the fundamental group of \(Y\) based at the 0-cell \(n\). The group is returned as a finitely presented group. If \(n\) is not specified then it is set \(n=1\). The algorithm requires a discrete vector field on \(Y\). If \(Y\) does not initially have a discrete vector field then one is constructed.

-

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 

+

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 

 [Top of Book]  [Contents]   [Previous Chapter]   [Next Chapter] 
diff --git a/doc/chap34.html b/doc/chap34.html index 86d20d0c..24be535c 100644 --- a/doc/chap34.html +++ b/doc/chap34.html @@ -38,14 +38,16 @@
  34.2-2 IdentityArrow
  34.2-3 InitialArrow
  34.2-4 TerminalArrow -
  34.2-5 Source -
  34.2-6 Target -
  34.2-7 CategoryName -
  34.2-8 CompositionEqualityAdditionMinus -
  34.2-9 Object -
  34.2-10 Mapping -
  34.2-11 IsCategoryObject -
  34.2-12 IsCategoryArrow +
  34.2-5 HasInitialObject +
  34.2-6 HasTerminalObject +
  34.2-7 Source +
  34.2-8 Target +
  34.2-9 CategoryName +
  34.2-10 CompositionEqualityAdditionMinus +
  34.2-11 Object +
  34.2-12 Mapping +
  34.2-13 IsCategoryObject +
  34.2-14 IsCategoryArrow
@@ -153,9 +155,27 @@
34.2-4 TerminalArrow

Examples: 1 

+

+ +
34.2-5 HasInitialObject
+ +
‣ HasInitialObject( Name )( function )
+

Inputs the name of a category and returns true or false depending on whether the category has an initial object.

+ +

Examples: 1 

+ +

+ +
34.2-6 HasTerminalObject
+ +
‣ HasTerminalObject( Name )( function )
+

Inputs the name of a category and returns true or false depending on whether the category has a terminal object.

+ +

Examples:

+

-
34.2-5 Source
+
34.2-7 Source
‣ Source( f )( function )

Inputs an arrow f in some category, and returns its source.

@@ -164,7 +184,7 @@
34.2-5 Source

-
34.2-6 Target
+
34.2-8 Target
‣ Target( f )( function )

Inputs an arrow f in some category, and returns its target.

@@ -173,7 +193,7 @@
34.2-6 Target

-
34.2-7 CategoryName
+
34.2-9 CategoryName
‣ CategoryName( X )( function )

Inputs an object or arrow X in some category, and returns the name of the category.

@@ -182,7 +202,7 @@
34.2-7 CategoryName

-
34.2-8 CompositionEqualityAdditionMinus
+
34.2-10 CompositionEqualityAdditionMinus
‣ CompositionEqualityAdditionMinus( global variable )

Composition of suitable arrows f,g is given by f*g when the source of f equals the target of g. (Warning: this differes to the standard GAP convention.)

@@ -195,7 +215,7 @@
34.2-8 CompositionEqualityAdditionMinus

-
34.2-9 Object
+
34.2-11 Object
‣ Object( X )( function )

Inputs an object X in some category, and returns the GAP structure Y such that X=CategoricalEnrichment(Y,CategoryName(X)).

@@ -204,7 +224,7 @@
34.2-9 Object

-
34.2-10 Mapping
+
34.2-12 Mapping
‣ Mapping( X )( function )

Inputs an arrow f in some category, and returns the GAP structure Y such that f=CategoricalEnrichment(Y,CategoryName(X)).

@@ -213,7 +233,7 @@
34.2-10 Mapping

-
34.2-11 IsCategoryObject
+
34.2-13 IsCategoryObject
‣ IsCategoryObject( X )( function )

Inputs X and returns true if X is an object in some category.

@@ -222,7 +242,7 @@
34.2-11 IsCategoryObject

-
34.2-12 IsCategoryArrow
+
34.2-14 IsCategoryArrow
‣ IsCategoryArrow( X )( function )

Inputs X and returns true if X is an arrow in some category.

diff --git a/doc/chap34.txt b/doc/chap34.txt index a7d1bd2e..738f4082 100644 --- a/doc/chap34.txt +++ b/doc/chap34.txt @@ -115,7 +115,25 @@ Examples: 1 (../www/SideLinks/About/aboutAbelianCategories.html)  - 34.2-5 Source + 34.2-5 HasInitialObject + + HasInitialObject( Name )  function + + Inputs the name of a category and returns true or false depending on whether + the category has an initial object. + + Examples: 1 (../www/SideLinks/About/aboutAbelianCategories.html)  + + 34.2-6 HasTerminalObject + + HasTerminalObject( Name )  function + + Inputs the name of a category and returns true or false depending on whether + the category has a terminal object. + + Examples: + + 34.2-7 Source Source( f )  function @@ -131,7 +149,7 @@ (../www/SideLinks/About/aboutFunctorial.html) , 11 (../www/SideLinks/About/aboutLieCovers.html)  - 34.2-6 Target + 34.2-8 Target Target( f )  function @@ -144,7 +162,7 @@ (../www/SideLinks/About/aboutCoveringSpaces.html) , 8 (../www/SideLinks/About/aboutCoverinSpaces.html)  - 34.2-7 CategoryName + 34.2-9 CategoryName CategoryName( X )  function @@ -153,7 +171,7 @@ Examples: 1 (../www/SideLinks/About/aboutAbelianCategories.html)  - 34.2-8 CompositionEqualityAdditionMinus + 34.2-10 CompositionEqualityAdditionMinus CompositionEqualityAdditionMinus  global variable @@ -168,7 +186,7 @@ Examples: - 34.2-9 Object + 34.2-11 Object Object( X )  function @@ -178,7 +196,7 @@ Examples: 1 (../tutorial/chap10.html) , 2 (../www/SideLinks/About/aboutAbelianCategories.html)  - 34.2-10 Mapping + 34.2-12 Mapping Mapping( X )  function @@ -190,7 +208,7 @@ (../www/SideLinks/About/aboutCoefficientSequence.html) , 5 (../www/SideLinks/About/aboutGouter.html)  - 34.2-11 IsCategoryObject + 34.2-13 IsCategoryObject IsCategoryObject( X )  function @@ -198,7 +216,7 @@ Examples: - 34.2-12 IsCategoryArrow + 34.2-14 IsCategoryArrow IsCategoryArrow( X )  function diff --git a/doc/chap34_mj.html b/doc/chap34_mj.html index cbf39336..18fae7b2 100644 --- a/doc/chap34_mj.html +++ b/doc/chap34_mj.html @@ -41,14 +41,16 @@
  34.2-2 IdentityArrow
  34.2-3 InitialArrow
  34.2-4 TerminalArrow -
  34.2-5 Source -
  34.2-6 Target -
  34.2-7 CategoryName -
  34.2-8 CompositionEqualityAdditionMinus -
  34.2-9 Object -
  34.2-10 Mapping -
  34.2-11 IsCategoryObject -
  34.2-12 IsCategoryArrow +
  34.2-5 HasInitialObject +
  34.2-6 HasTerminalObject +
  34.2-7 Source +
  34.2-8 Target +
  34.2-9 CategoryName +
  34.2-10 CompositionEqualityAdditionMinus +
  34.2-11 Object +
  34.2-12 Mapping +
  34.2-13 IsCategoryObject +
  34.2-14 IsCategoryArrow @@ -156,9 +158,27 @@
34.2-4 TerminalArrow

Examples: 1 

+

+ +
34.2-5 HasInitialObject
+ +
‣ HasInitialObject( Name )( function )
+

Inputs the name of a category and returns true or false depending on whether the category has an initial object.

+ +

Examples: 1 

+ +

+ +
34.2-6 HasTerminalObject
+ +
‣ HasTerminalObject( Name )( function )
+

Inputs the name of a category and returns true or false depending on whether the category has a terminal object.

+ +

Examples:

+

-
34.2-5 Source
+
34.2-7 Source
‣ Source( f )( function )

Inputs an arrow \(f\) in some category, and returns its source.

@@ -167,7 +187,7 @@
34.2-5 Source

-
34.2-6 Target
+
34.2-8 Target
‣ Target( f )( function )

Inputs an arrow \(f\) in some category, and returns its target.

@@ -176,7 +196,7 @@
34.2-6 Target

-
34.2-7 CategoryName
+
34.2-9 CategoryName
‣ CategoryName( X )( function )

Inputs an object or arrow \(X\) in some category, and returns the name of the category.

@@ -185,7 +205,7 @@
34.2-7 CategoryName

-
34.2-8 CompositionEqualityAdditionMinus
+
34.2-10 CompositionEqualityAdditionMinus
‣ CompositionEqualityAdditionMinus( global variable )

Composition of suitable arrows \(f,g\) is given by \(f*g\) when the source of \(f\) equals the target of \(g\). (Warning: this differes to the standard GAP convention.)

@@ -198,7 +218,7 @@
34.2-8 CompositionEqualityAdditionMinus

-
34.2-9 Object
+
34.2-11 Object
‣ Object( X )( function )

Inputs an object \(X\) in some category, and returns the GAP structure \(Y\) such that \(X=CategoricalEnrichment(Y,CategoryName(X))\).

@@ -207,7 +227,7 @@
34.2-9 Object

-
34.2-10 Mapping
+
34.2-12 Mapping
‣ Mapping( X )( function )

Inputs an arrow \(f\) in some category, and returns the GAP structure \(Y\) such that \(f=CategoricalEnrichment(Y,CategoryName(X))\).

@@ -216,7 +236,7 @@
34.2-10 Mapping

-
34.2-11 IsCategoryObject
+
34.2-13 IsCategoryObject
‣ IsCategoryObject( X )( function )

Inputs \(X\) and returns true if \(X\) is an object in some category.

@@ -225,7 +245,7 @@
34.2-11 IsCategoryObject

-
34.2-12 IsCategoryArrow
+
34.2-14 IsCategoryArrow
‣ IsCategoryArrow( X )( function )

Inputs \(X\) and returns true if \(X\) is an arrow in some category.

diff --git a/doc/chap40.html b/doc/chap40.html index a6f3dd75..512988aa 100644 --- a/doc/chap40.html +++ b/doc/chap40.html @@ -33,7 +33,7 @@

40 HAP variables that are not yet documented40.1  

-

2CoreducedChainComplex    Examples:

AbelianGOuterGroupToCatOneGroup    Examples:

AbelianInvariantsToTorsionCoefficients    Examples:

AcyclicSubcomplexOfPureCubicalComplex    Examples: 1 

AddFirst    Examples:

AdjointGroupOfQuandle    Examples: 1 

AlgebraicReduction_alt    Examples:

AppendFreeWord    Examples:

ArcDiagramToTubularSurface    Examples:

ArcPresentation    Examples: 1 , 2 , 3 , 4 

ArcPresentationToKnottedOneComplex    Examples:

AreIsoclinic    Examples:

ArrayIterateBreak    Examples:

ArrayValueKD    Examples:

AsWordInSL2Z    Examples:

AutomorphismGroupQuandleAsPerm_nonconnected    Examples:

AverageInnerProduct    Examples:

BarCodeOfFilteredPureCubicalComplex    Examples:

BarCodeOfSymmetricMatrix    Examples:

BarComplexOfMonoid    Examples: 1 

BarycentricallySimplifiedComplex    Examples: 1 

BarycentricallySubdivideCell    Examples:

BettinumbersOfPureCubicalComplex_dim_2    Examples:

BocksteinHomology    Examples:

BogomolovMultiplier_viaTensorSquare    Examples:

BoundariesOfFilteredChainComplex    Examples:

BoundaryOfPureComplex    Examples: 1 

BoundaryOfPureRegularCWComplex    Examples: 1 

BoundaryOfRegularCWCell    Examples:

BoundaryPairOfPureRegularCWComplex    Examples:

BoundingPureComplex    Examples:

CR_ChainMapFromCocycle    Examples:

CR_CocyclesAndCoboundaries    Examples:

CR_IntegralClassToCocycle    Examples:

CR_IntegralCocycleToClass    Examples:

CR_IntegralCohomology    Examples:

CR_IntegralCycleToClass    Examples:

CWMap2ChainMap    Examples:

CWSubcomplexToRegularCWMap    Examples: 1 

CanonicalRightCountableCosetElement    Examples:

CatOneGroupByCrossedModule    Examples:

CatOneGroupsByGroup    Examples:

CcElement    Examples:

Cedric_CheckThirdAxiomRow    Examples:

Cedric_ConjugateQuandleElement    Examples:

Cedric_FromAutGeReToAutQe    Examples:

Cedric_IsHomomorphism    Examples:

Cedric_Permute    Examples:

Cedric_Quandle1    Examples:

Cedric_Quandle2    Examples:

Cedric_Quandle3    Examples:

Cedric_Quandle4    Examples:

Cedric_Quandle5    Examples:

Cedric_Quandle6    Examples:

CellComplexBoundaryCheck    Examples:

ChainComplexEquivalenceOfRegularCWComplex    Examples: 1 

ChainComplexHomeomorphismEquivalenceOfRegularCWComplex    Examples:

ChainComplexOfCubicalComplex    Examples:

ChainComplexOfCubicalPair    Examples:

ChainComplexOfRegularCWComplexWithVectorField    Examples:

ChainComplexOfSimplicialComplex    Examples:

ChainComplexOfSimplicialPair    Examples:

ChainComplexOfUniversalCover    Examples: 1 , 2 , 3 , 4 

ChainComplexToSparseChainComplex    Examples:

ChainComplexWithChainHomotopy    Examples:

ChainMapOfCubicalPairs    Examples:

ChainMapOfRegularCWMap    Examples:

ChildRestart    Examples:

ClosureCWCell    Examples:

CoClass    Examples:

CocriticalCellsOfRegularCWComplex    Examples:

CocyclicHadamardMatrices    Examples: 1 

CocyclicMatrices    Examples:

CohomologicalData    Examples: 1 

CohomologyHomomorphism    Examples: 1 , 2 

CohomologyHomomorphismOfRepresentation    Examples:

CohomologyModule_AsAutModule    Examples:

CohomologyModule_Gmap    Examples:

CohomologyRingOfSimplicialComplex    Examples:

CohomologySimplicialFreeAbelianGroup    Examples:

CombinationDisjointSets    Examples:

CommonEndomorphisms    Examples:

ComplementOfPureComplex    Examples: 1 

ComplementaryBasis    Examples:

ComposeCWMaps    Examples:

CompositionOfFpGModuleHomomorphisms    Examples:

CompositionSeriesOfFpGModule    Examples:

ConcentricallyFilteredPureCubicalComplex    Examples: 1 

CongruenceSubgroup    Examples: 1 , 2 

ConjugateSL2ZGroup    Examples:

ConnectingCohomologyHomomorphism    Examples: 1 , 2 

ContractArray    Examples:

ContractCubicalComplex_dim2    Examples:

ContractCubicalComplex_dim3    Examples:

ContractMatrix    Examples:

ContractPermArray    Examples:

ContractPermMatrix    Examples:

ContractPureComplex    Examples:

ContractSimplicialComplex    Examples:

ContractSimplicialComplex_alt    Examples:

ContractedFilteredPureCubicalComplex    Examples: 1 

ContractedFilteredRegularCWComplex    Examples:

ContractedRegularCWComplex    Examples:

ContractibleSL2ZComplex    Examples:

ContractibleSL2ZComplex_alt    Examples:

ContractibleSubArray    Examples:

ContractibleSubMatrix    Examples:

ContractibleSubcomplexOfPureCubicalComplex    Examples: 1 

ConvertTorsionComplexToGcomplex    Examples:

CosetsQuandle    Examples:

CountingCellsOfBaryCentricSubdivision    Examples:

CountingNumberOfCellsInBaryCentricSubdivision    Examples:

CoxeterComplex_alt    Examples: 1 

CoxeterDiagramMatCoxeterGroup    Examples:

CoxeterWythoffComplex    Examples:

CreateCoxeterMatrix    Examples: 1 

CriticalBoundaryCells    Examples: 1 

CropPureComplex    Examples:

CrossedInvariant    Examples:

CrossedModuleByAutomorphismGroup    Examples:

CrossedModuleByCatOneGroup    Examples:

CrossedModuleByNormalSubgroup    Examples: 1 

CrystCubicalTiling    Examples:

CrystFinitePartOfMatrix    Examples:

CrystGFullBasis    Examples: 1 , 2 

CrystGcomplex    Examples: 1 , 2 

CrystMatrix    Examples:

CrystTranslationMatrixToVector    Examples:

CrystallographicComplex    Examples:

CubicalToPermutahedralArray    Examples:

CupProductMatrix    Examples:

CupProductOfRegularCWComplex    Examples: 1 

CupProductOfRegularCWComplex_alt    Examples: 1 

CuspidalCohomologyHomomorphism    Examples:

CyclesOfFilteredChainComplex    Examples:

DavisComplex    Examples: 1 , 2 , 3 , 4 

DeformationRetract    Examples:

DensityMat    Examples:

DerivedGroupOfQuandle    Examples: 1 

DiagonalChainMap    Examples:

DijkgraafWittenInvariant    Examples: 1 

DirectProductOfGroupHomomorphisms    Examples:

DirectProductOfRegularCWComplexes    Examples:

DirectProductOfRegularCWComplexesLazy    Examples:

DirectProductOfSimplicialComplexes    Examples:

DisplayCSVknotFile    Examples:

DisplayVectorField    Examples:

E1CohomologyPage    Examples:

E1HomologyPage    Examples:

EilenbergMacLaneSimplicialFreeAbelianGroup    Examples:

ElementsLazy    Examples:

EquivariantCWComplexToRegularCWComplex    Examples: 1 , 2 , 3 , 4 

EquivariantCWComplexToRegularCWMap    Examples: 1 , 2 , 3 

EquivariantCWComplexToResolution    Examples:

ExcisedPureCubicalPair_dim_2    Examples:

ExtractTorsionSubcomplex    Examples:

FactorizationNParts    Examples:

FilteredChainComplexToFilteredSparseChainComplex    Examples:

FilteredCubicalComplexToFilteredRegularCWComplex    Examples: 1 

FilteredPureCubicalComplexToCubicalComplex    Examples: 1 

FiltrationTermOfGraph    Examples:

FiltrationTermOfPureCubicalComplex    Examples:

FiltrationTermOfRegularCWComplex    Examples:

FirstHomologyCoveringCokernels    Examples: 1 , 2 

FirstHomologySimplicialTwoComplex    Examples:

FourthHomotopyGroupOfDoubleSuspensionB    Examples:

Fp2PcpAbelianGroupHomomorphism    Examples:

FpGModuleSection    Examples:

FreeZGResolution    Examples:

FundamentalGroupOfRegularCWComplex    Examples: 1 

FundamentalGroupOfRegularCWMap    Examples:

FundamentalGroupSimplicialTwoComplex    Examples:

FundamentalMultiplesOfStiefelWhitneyClasses    Examples:

GChainComplex    Examples: 1 

GModuleAsCatOneGroup    Examples:

GammaSubgroupInSL3Z    Examples:

GaussCodeOfPureCubicalKnot    Examples: 1 , 2 , 3 , 4 

GetTorsionPowerSubcomplex    Examples:

GetTorsionSubcomplex    Examples:

GraphOfRegularCWComplex    Examples:

GraphOfResolutionsTest    Examples:

GraphOfResolutionsToGroups    Examples:

GroupHomomorphismToMatrix    Examples:

HAPCocontractRegularCWComplex    Examples:

HAPContractFilteredRegularCWComplex    Examples:

HAPContractRegularCWComplex    Examples:

HAPContractRegularCWComplex_Alt    Examples:

HAPPRIME_Algebra2Polynomial    Examples:

HAPPRIME_CohomologyRingWithoutResolution    Examples:

HAPPRIME_CombineIndeterminateMaps    Examples:

HAPPRIME_GradedAlgebraPresentationAvoidingIndeterminates    Examples:

HAPPRIME_LHSSpectralSequence    Examples:

HAPPRIME_MakeEliminationOrdering    Examples:

HAPPRIME_MapPolynomialIndeterminates    Examples:

HAPPRIME_Polynomial2Algebra    Examples:

HAPPRIME_RingHomomorphismsAreComposable    Examples:

HAPPRIME_SModule    Examples:

HAPPRIME_SingularGroebnerBasis    Examples:

HAPPRIME_SingularReducedGroebnerBasis    Examples:

HAPPRIME_SwitchGradedAlgebraRing    Examples:

HAPPRIME_SwitchPolynomialIndeterminates    Examples:

HAPPRIME_VersionWithSVN    Examples:

HAPRegularCWComplex    Examples:

HAPRegularCWPolytope    Examples:

HAPRemoveCellFromRegularCWComplex    Examples:

HAPRemoveVectorField    Examples:

HAPRingModIdeal    Examples:

HAPRingModIdealObj    Examples:

HAPTietzeReduction_Inf    Examples:

HAPTietzeReduction_OneLevel    Examples:

HAPTietzeReduction_OneStep    Examples:

HAP_4x4MatTo2x2Mat    Examples:

HAP_AddGenerator    Examples:

HAP_AllHomomorphisms    Examples:

HAP_AppendTo    Examples:

HAP_AssociahedronBoundaries    Examples:

HAP_AssociahedronCells    Examples:

HAP_BaryCentricSubdivisionGComplex    Examples:

HAP_BaryCentricSubdivisionRegularCWComplex    Examples:

HAP_Binlisttoint    Examples:

HAP_ChainComplexToEquivariantChainComplex    Examples:

HAP_CocyclesAndCoboundaries    Examples:

HAP_CongruenceSubgroupGamma0    Examples: 1 

HAP_CongruenceSubgroupGamma0Ideal    Examples:

HAP_ConjugatedCongruenceSubgroup    Examples:

HAP_ConjugatedCongruenceSubgroupGamma0    Examples:

HAP_CriticalCellsDirected    Examples:

HAP_CupProductOfPresentation    Examples:

HAP_CupProductOfSimplicialComplex    Examples:

HAP_DisplayPlanarTree    Examples:

HAP_DisplayVectorField    Examples:

HAP_ElementsSL2Zfn    Examples:

HAP_FunctorialModPCohomologyRing    Examples:

HAP_GenericSL2OSubgroup    Examples:

HAP_GenericSL2ZConjugatedSubgroup    Examples:

HAP_GenericSL2ZSubgroup    Examples:

HAP_HomToIntModP_ChainComplex    Examples:

HAP_HomToIntModP_ChainMap    Examples:

HAP_HomToIntModP_CochainComplex    Examples:

HAP_HomToIntModP_CochainMap    Examples:

HAP_HomeoLinkingForm    Examples:

HAP_Hurewicz1Cycles    Examples:

HAP_IntegralClassToCocycle    Examples:

HAP_IntegralCocycleToClass    Examples:

HAP_IntegralCohomology    Examples:

HAP_KK_AddCell    Examples:

HAP_KnotGroupInv    Examples:

HAP_MyIsBieberbachFpGroup    Examples:

HAP_MyIsFiniteFpGroup    Examples:

HAP_MyIsInfiniteFpGroup    Examples:

HAP_PHI    Examples:

HAP_PermBinlisttoint    Examples:

HAP_PlanarBinaryTrees    Examples:

HAP_PlanarTreeGraft    Examples:

HAP_PlanarTreeJoin    Examples:

HAP_PlanarTreeLeaves    Examples:

HAP_PlanarTreeRemovableEdge    Examples:

HAP_PlanarTreeRemoveEdge    Examples:

HAP_PrimePartModified    Examples:

HAP_PrincipalCongruenceSubgroup    Examples:

HAP_PrincipalCongruenceSubgroupIdeal    Examples:

HAP_PrintTo    Examples:

HAP_PureComplexSubcomplex    Examples:

HAP_PureCubicalPairToCWMap    Examples:

HAP_ResolutionAbelianGroupFromInvariants    Examples:

HAP_RightTransversalSL2ZSubgroups    Examples:

HAP_SL2OSubgroupTree_slow    Examples:

HAP_SL2SubgroupTree    Examples:

HAP_SL2TreeDisplay    Examples:

HAP_SL2ZSubgroupTree_fast    Examples:

HAP_SL2ZSubgroupTree_slow    Examples:

HAP_Sequence2Boundaries    Examples:

HAP_SimplicialPairToCWMap    Examples:

HAP_SimplicialProjectivePlane    Examples:

HAP_SimplicialTorus    Examples:

HAP_SimplifiedGaussCode    Examples:

HAP_StiefelWhitney    Examples:

HAP_SylowSubgroups    Examples:

HAP_Tensor    Examples:

HAP_TransversalCongruenceSubgroups    Examples:

HAP_TransversalCongruenceSubgroupsIdeal    Examples:

HAP_TransversalCongruenceSubgroupsIdeal_alt    Examples:

HAP_TransversalGamma0SubgroupsIdeal    Examples:

HAP_Triangulation    Examples:

HAP_TzPair    Examples:

HAP_WedgeSumOfSimplicialComplexes    Examples:

HAP_bockstein    Examples:

HAP_chain_bockstein    Examples:

HAP_coho_isoms    Examples:

HAP_nxnMatTo2nx2nMat    Examples:

HadamardGraph    Examples:

HapExample    Examples:

HapFile    Examples: 1 , 2 , 3 , 4 

HasTrivialPostnikovInvariant    Examples:

HeckeOperator    Examples:

HeckeOperatorWeight2    Examples:

HenonOrbit    Examples: 1 

HomToGModule_hom    Examples:

HomToInt_ChainComplex    Examples:

HomToInt_ChainMap    Examples:

HomToInt_CochainComplex    Examples:

HomToModPModule    Examples: 1 

HomogeneousPolynomials    Examples:

HomogeneousPolynomials_Bianchi    Examples:

HomologicalGroupDecomposition    Examples: 1 

HomologyOfPureCubicalComplex    Examples:

HomologyPbs    Examples:

HomologySimplicialFreeAbelianGroup    Examples:

HomomorphismAsMatrix    Examples:

HomotopyCatOneGroup    Examples:

HomotopyCrossedModule    Examples:

HomotopyEquivalentLargerSubArray    Examples:

HomotopyEquivalentLargerSubArray3D    Examples:

HomotopyEquivalentLargerSubMatrix    Examples:

HomotopyEquivalentLargerSubPermArray    Examples:

HomotopyEquivalentLargerSubPermArray3D    Examples:

HomotopyEquivalentLargerSubPermMatrix    Examples:

HomotopyEquivalentMaximalPureSubcomplex    Examples:

HomotopyEquivalentMinimalPureSubcomplex    Examples:

HomotopyEquivalentSmallerSubArray    Examples:

HomotopyEquivalentSmallerSubArray3D    Examples:

HomotopyEquivalentSmallerSubMatrix    Examples:

HomotopyEquivalentSmallerSubPermArray    Examples:

HomotopyEquivalentSmallerSubPermArray3D    Examples:

HomotopyEquivalentSmallerSubPermMatrix    Examples:

HomotopyLowerCentralSeries    Examples:

HomotopyLowerCentralSeriesOfCrossedModule    Examples:

HomotopyTruncation    Examples:

HopfSatohSurface    Examples: 1 , 2 

HybridSubdivision    Examples:

IdCatOneGroup    Examples: 1 

IdCrossedModule    Examples:

IdQuasiCatOneGroup    Examples:

IdQuasiCrossedModule    Examples:

IdentifyKnot    Examples: 1 

IdentityAmongRelators    Examples: 1 , 2 

ImageOfGOuterGroupHomomorphism    Examples: 1 , 2 

ImageOfMap    Examples:

InducedSteenrodHomomorphisms    Examples:

IntegerSimplicialComplex    Examples: 1 

IntegralCellularHomology    Examples:

IntegralCohomology    Examples:

IntegralCohomologyOfCochainComplex    Examples:

IntegralHomology    Examples: 1 

IntegralHomologyOfChainComplex    Examples:

IntersectionCWSubcomplex    Examples:

IsClosedManifold    Examples: 1 

IsContractibleCube_higherdims    Examples:

IsCrystSameOrbit    Examples:

IsCrystSufficientLattice    Examples:

IsHadamardMatrix    Examples:

IsIntList    Examples:

IsIsomorphismOfAbelianFpGroups    Examples: 1 

IsMetricMatrix    Examples:

IsPeriodicSpaceGroup    Examples: 1 

IsPureComplex    Examples:

IsPureRegularCWComplex    Examples:

IsRigid    Examples: 1 

IsRigidOnRight    Examples:

IsSphericalCoxeterGroup    Examples:

IsoclinismClasses    Examples: 1 , 2 

IsomorphismCatOneGroups    Examples: 1 

IsomorphismCrossedModules    Examples:

KernelOfGOuterGroupHomomorphism    Examples: 1 , 2 

KernelOfMap    Examples:

KernelWG    Examples:

KinkArc2Presentation    Examples:

KnotComplement    Examples: 1 , 2 , 3 

KnotComplementWithBoundary    Examples: 1 , 2 , 3 

LazyList    Examples:

LefschetzNumberOfChainMap    Examples:

Lfunction    Examples:

LiftColouredSurface    Examples:

LiftedRegularCWMap    Examples:

LinearHomomorphismsZZPersistenceMat    Examples:

LinkingForm    Examples: 1 

LinkingFormHomeomorphismInvariant    Examples: 1 

LinkingFormHomotopyInvariant    Examples: 1 

ListsOfCellsToRegularCWComplex    Examples:

LowDimensionalCupProduct    Examples: 1 

MakeHAPprimeDoc    Examples:

ManifoldType    Examples: 1 

Mapper    Examples: 1 

Mapper_alt    Examples:

MatrixSize    Examples:

MaximalSimplicesOfSimplicialComplex    Examples: 1 

MaximalSphericalCoxeterSubgroupsFromAbove    Examples:

MinimizeRingRelations    Examples:

Mod2SteenrodAlgebra    Examples: 1 

ModPCohomologyRing_alt    Examples:

ModPCohomologyRing_part_1    Examples:

ModPCohomologyRing_part_2    Examples:

ModPRingGeneratorsAlt    Examples:

ModPSteenrodAlgebra    Examples: 1 , 2 

ModularCohomology    Examples:

ModularEquivariantChainMap    Examples:

ModularHomology    Examples:

Nil3TensorSquare    Examples:

NonFreeResolutionFiniteSubgroup    Examples:

NonManifoldVertices    Examples:

NonRegularCWBoundary    Examples:

NonabelianSymmetricKernel_alt    Examples: 1 

NonabelianSymmetricSquare_inf    Examples:

NonabelianTensorProduct_Inf    Examples:

NonabelianTensorProduct_alt    Examples:

NonabelianTensorSquareAsCatOneGroup    Examples:

NonabelianTensorSquareAsCrossedModule    Examples:

NonabelianTensorSquare_inf    Examples:

NoncrossingPartitionsLatticeDisplay    Examples: 1 

NullspaceSparseMatDestructive    Examples:

NumberConnectedQuandles    Examples:

NumberGeneratorsOfGroupHomology    Examples:

NumberOfCrossingsInArc2Presentation    Examples:

NumberOfHomomorphisms_connected    Examples:

NumberOfHomomorphisms_groups    Examples:

NumberOfPrimeKnots    Examples: 1 , 2 

NumberSmallCatOneGroups    Examples:

NumberSmallCrossedModules    Examples:

NumberSmallQuasiCatOneGroups    Examples:

NumberSmallQuasiCrossedModules    Examples:

OppositeGroup    Examples:

OrthogonalizeBasisByAverageInnerProduct    Examples:

PCentre    Examples:

PSubgroupGChainComplex    Examples:

PSubgroupSimplicialComplex    Examples:

PUpperCentralSeries    Examples:

PartialIsoclinismClasses    Examples: 1 

PartsOfQuadraticInteger    Examples:

PathComponentOfPureComplex    Examples: 1 

PathComponentsCWSubcomplex    Examples:

PathComponentsOfSimplicialComplex_alt    Examples:

PathObjectForChainComplex    Examples: 1 

PermutahedralComplexToRegularCWComplex    Examples: 1 

PermutahedralToCubicalArray    Examples:

PersistentBettiNumbersViaContractions    Examples:

PersistentHomologyOfCrossedModule    Examples:

PersistentHomologyOfFilteredPureCubicalComplex_alt    Examples:

PersistentHomologyOfFilteredSparseChainComplex    Examples: 1 , 2 

PersistentHomologyOfPureCubicalComplex_Alt    Examples:

PersistentHomologyOfQuotientGroupSeries_Int    Examples:

PiZeroOfRegularCWComplex    Examples:

PoincareBipyramidCWComplex    Examples: 1 

PoincareCubeCWComplex    Examples: 1 

PoincareCubeCWComplexNS    Examples: 1 

PoincareDodecahedronCWComplex    Examples: 1 

PoincareOctahedronCWComplex    Examples: 1 

PoincarePrismCWComplex    Examples: 1 

PoincareSeriesApproximation    Examples:

PoincareSeries_alt    Examples:

PolymakeFaceLattice    Examples:

PolytopalRepresentationComplex    Examples:

PrankAlt    Examples:

PresentationOfResolution_alt    Examples:

PrimePartDerivedFunctorHomomorphism    Examples:

PrimePartDerivedFunctorViaSubgroupChain    Examples:

PrimePartDerivedTwistedFunctor    Examples:

PrintAlgebraWordAsPolynomial    Examples:

PrintTorsionSubcomplex    Examples:

PureComplex    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 

PureCubicalComplexToCubicalComplex    Examples: 1 , 2 

PureCubicalLink    Examples: 1 , 2 

PushoutOfFpGroups    Examples:

QuadraticCharacter    Examples:

QuadraticNumberField    Examples: 1 

QuandleIsomorphismRepresentatives    Examples:

QuotientByTorsionSubcomplex    Examples:

QuotientChainMap    Examples:

QuotientGroup    Examples:

QuotientQuasiIsomorph    Examples:

RadicalSeriesOfResolution    Examples:

RandomArc2Presentation    Examples:

RandomCellOfPureComplex    Examples:

ReadLinkImageAsGaussCode    Examples: 1 

ReadMatrixAsPureCubicalComplex    Examples:

ReduceGenerators    Examples:

ReduceGenerators_alt    Examples:

ReflectedCubicalKnot    Examples: 1 , 2 , 3 , 4 

RegularCWAssociahedron    Examples:

RegularCWComplexComplement    Examples: 1 

RegularCWComplexWithRemovedCell    Examples: 1 

RegularCWComplex_AttachCellDestructive    Examples: 1 

RegularCWCube    Examples:

RegularCWMapToCWSubcomplex    Examples:

RegularCWOrbitPolytope    Examples:

RegularCWPermutahedron    Examples:

RegularCWPolygon    Examples:

RegularCWSimplex    Examples:

RelativeCentralQuotientSpaceGroup    Examples:

RelativeGroupHomology    Examples:

RelativeRightTransversal    Examples:

RemoveStar    Examples:

ResolutionAbelianGroup_alt    Examples:

ResolutionAbelianPcpGroup    Examples:

ResolutionAffineCrystGroup    Examples:

ResolutionBoundaryOfWordOnRight    Examples:

ResolutionDirectProductLazy    Examples:

ResolutionFiniteCyclicGroup    Examples:

ResolutionGL2QuadraticIntegers    Examples:

ResolutionGL3QuadraticIntegers    Examples:

ResolutionGenericGroup    Examples:

ResolutionInfiniteCyclicGroup    Examples:

ResolutionPGL2QuadraticIntegers    Examples:

ResolutionPGL3QuadraticIntegers    Examples:

ResolutionPSL2QuadraticIntegers    Examples: 1 

ResolutionPrimePowerGroupSparse    Examples:

ResolutionSL2QuadraticIntegers    Examples: 1 

ResolutionSL2ZConjugated    Examples:

ResolutionSL2Z_alt    Examples:

ResolutionSpaceGroup    Examples: 1 

ResolutionToEquivariantCWComplex    Examples:

ResolutionToResolutionOfFpGroup    Examples: 1 

SL2QuadraticIntegers    Examples: 1 

SL2ZResolution    Examples:

SL2ZResolution_alt    Examples:

SL2ZTree    Examples:

SL2ZmElementsDecomposition    Examples:

SequentialRegularCWComplexComplement    Examples:

SignatureOfSymmetricMatrix    Examples: 1 

SignedPermutationGroup    Examples: 1 

SimplicesToSimplicialComplex    Examples: 1 , 2 , 3 , 4 

SimplicialComplexToRegularCWComplex_alt    Examples:

SimplicialK3Surface    Examples: 1 

SimplicialNerveOfFilteredGraph    Examples: 1 , 2 

SimplicialNerveOfTwoComplex    Examples:

SimplifiedQuandlePresentation    Examples:

SimplifiedRegularCWComplex    Examples: 1 

SimplifiedSparseChainComplex    Examples:

SmallCatOneGroup    Examples: 1 

SmallCrossedModule    Examples:

SmallQuasiCatOneGroup    Examples:

SmallQuasiCrossedModule    Examples:

SmoothedFpGroup    Examples:

SparseChainComplexOfCubicalComplex    Examples:

SparseChainComplexOfCubicalPair    Examples:

SparseChainComplexOfFilteredRegularCWComplex    Examples:

SparseChainComplexOfRegularCWComplexWithVectorField    Examples:

SparseChainComplexOfSimplicialComplex    Examples:

SparseChainComplexToChainComplex    Examples:

SparseChainMapOfCubicalPairs    Examples:

SparseFilteredChainComplexOfFilteredCubicalComplex    Examples:

SparseFilteredChainComplexOfFilteredSimplicialComplex    Examples: 1 , 2 

SparseMattoMat    Examples: 1 

SparseRowReduce    Examples:

SphericalKnotComplement    Examples: 1 

Spin    Examples:

SpunAboutHyperplane    Examples:

SpunKnotComplement    Examples: 1 

SpunLinkComplement    Examples:

StrongGeneratorsOfDerivedSubgroup    Examples:

StrongGeneratorsOfDerivedSubgroup_alt    Examples:

StructuralCopyOfFilteredRegularCWComplex    Examples:

SubQuasiIsomorph    Examples:

SubdivideCell    Examples:

Suspension_alt    Examples:

SylowSubgroupOfCatOneGroup    Examples:

SymmetricCentre    Examples:

SymmetricCommutativityGroup    Examples:

TensorNonFreeResolutionWithRationals    Examples:

TensorWithBurnsideRing    Examples: 1 , 2 

TensorWithComplexRepresentationRing    Examples: 1 , 2 

TensorWithComplexRepresentationRingOnRight    Examples:

TensorWithIntegersModPSparse    Examples:

TensorWithIntegersOverSubgroup    Examples: 1 , 2 , 3 , 4 

TensorWithIntegersSparse    Examples:

TensorWithModPModule    Examples: 1 

TestHapBook    Examples:

TestHapQuick    Examples:

ThickenedHEPureCubicalComplex    Examples:

ThickenedPureComplex    Examples: 1 

ThickenedPureCubicalComplex_dim2    Examples:

ThirdHomotopyGroupOfSuspensionB_alt    Examples: 1 

ThreeManifoldViaDehnSurgery    Examples: 1 

ThreeManifoldWithBoundary    Examples: 1 

TransferChainMap    Examples: 1 

TransferCochainMap    Examples: 1 

TranslationSubGroup    Examples:

TreeOfResolutionsToSL2Zcomplex    Examples:

TruncatedRegularCWComplex    Examples:

Tube    Examples:

TupleOrbitReps    Examples:

TupleOrbitReps_perm    Examples:

TwistedResolution    Examples:

UnboundedArrayAssign    Examples:

UnitBall    Examples:

UnitCubicalBall    Examples:

UnitPermutahedralBall    Examples:

UniversalBarCodeEval    Examples:

UniversalCover    Examples: 1 , 2 , 3 , 4 

VectorToCrystMatrix    Examples:

VectorsToOneSkeleton    Examples: 1 

VerticesOfRegularCWCell    Examples:

View3dPureComplex    Examples:

ViewArc2Presentation    Examples:

ViewPureComplex    Examples:

VirtuallySimplicialSubdivision    Examples:

WeakCommutativityGroup    Examples:

WirtingerGroup    Examples: 1 

WirtingerGroup_gc    Examples:

WordModP    Examples:

ZigZagContractedFilteredPureCubicalComplex    Examples:

ZigZagContractedPureComplex    Examples: 1 

Sq    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 

Category_Of_Groups    Examples: 1 

ElementsSL2Z    Examples:

HAP_knot_census    Examples:

PathComponentOfSimplicialComplex    Examples:

ResolutionSL2ZInvertedInteger    Examples:

ViewGraph    Examples:

AsFpGroup    Examples:

BarycentricSubdivision    Examples: 1 , 2 

Bockstein    Examples: 1 , 2 , 3 

CategoryArrow    Examples:

CategoryObject    Examples:

ClosedSurface    Examples: 1 

CoboundaryMatrix    Examples:

CoefficientsOfPoincareSeries    Examples:

CohomologyClass    Examples: 1 , 2 

CohomologyRing    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 

ComplexProjectiveSpace    Examples: 1 

CompositionRingHomomorphism    Examples:

ConnectedComponentsQuandle    Examples:

ConnectedSum    Examples: 1 , 2 

DegreeOfRepresentative    Examples:

Dimensions    Examples:

ExcisedPair    Examples:

FilteredRegularCWComplex    Examples: 1 

FundamentalGroupWithPathReps    Examples: 1 , 2 

GDerivedSubgroup    Examples:

GModuleAsGOuterGroup    Examples: 1 , 2 , 3 , 4 

GOuterGroupHomomorphism    Examples: 1 , 2 

GOuterGroupHomomorphism    Examples: 1 , 2 

GradedAlgebraPresentation    Examples:

GradedAlgebraPresentationNC    Examples:

HAPDerivationNC    Examples:

HAPRingHomomorphismByIndeterminateMap    Examples:

HAPRingReductionHomomorphism    Examples:

HAPRingReductionHomomorphism    Examples:

HAPRingToSubringHomomorphism    Examples:

HAPSubringToRingHomomorphism    Examples:

HAPSubringToRingHomomorphism    Examples:

HAPZeroRingHomomorphism    Examples:

HAP_EquivalenceClasses    Examples:

HomomorphismsImages    Examples:

ImageOfDerivation    Examples:

ImageOfRingHomomorphism    Examples:

IsAssociatedGradedRing    Examples:

KernelOfDerivation    Examples:

LowerGCentralSeries    Examples:

PathComponents    Examples: 1 

PersistentBettiNumbersAlt    Examples: 1 

PersistentHomology    Examples: 1 , 2 

PersistentHomology    Examples: 1 , 2 

PersistentHomology    Examples: 1 , 2 

PoincareSeriesAutoMem    Examples:

PoincareSeriesAutoMem    Examples:

PoincareSeriesAutoMemStop    Examples:

PolynomialToRModuleRep    Examples:

PreimageOfRingHomomorphism    Examples:

PureComplexMeet    Examples:

PureComplexRandomCell    Examples:

PureComplexSubcomplex    Examples:

Pushout    Examples:

QuadraticIdeal    Examples: 1 

ReduceIdeal    Examples:

ReducedPolynomialRingPresentation    Examples:

ReducedPolynomialRingPresentationMap    Examples:

RefinedColouring    Examples:

Resolution    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 

RightTransversal_alt    Examples:

RingOfIntegers    Examples: 1 

SingularPolynomialNormalForm    Examples:

SingularSetNormalFormIdeal    Examples:

SingularSetNormalFormIdealNC    Examples:

SparseChainComplexOfPair    Examples:

Sphere    Examples: 1 

Sq    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 

Standard2Cocycle    Examples:

Standard2Cocycle    Examples:

StandardNCocycle    Examples:

StandardNCocycle    Examples:

StarGraph    Examples:

SubspaceBasisRepsByDegree    Examples:

SubspaceDimensionDegree    Examples:

Suspension    Examples: 1 , 2 , 3 , 4 , 5 

TrivialGModuleAsGOuterGroup    Examples: 1 , 2 , 3 , 4 

VertexLink    Examples:

VertexStar    Examples:

WedgeSum    Examples: 1 

TensorProductOp    Examples:

Arity    Examples:

AssociatedNumberField    Examples:

AssociatedRing    Examples:

Base    Examples: 1 

BaseElement    Examples:

BaseRing    Examples:

Cocycle    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 

CoefficientModule    Examples:

CohomologicalPeriod    Examples: 1 

CoxeterMatrix    Examples: 1 

DerivationImages    Examples:

DerivationRelations    Examples:

DerivationRing    Examples:

Fibre    Examples:

FibreElement    Examples:

Generators    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 

GeneratorsOfPresentationIdeal    Examples:

GradedAlgebraPresentationFamily    Examples:

HAPDerivationFamily    Examples:

HAPPRIME_HilbertSeries    Examples:

HAPRingHomomorphismFamily    Examples:

HAP_MultiplicativeGenerators    Examples:

IdentityMap    Examples:

ImageGenerators    Examples:

ImagePolynomialRing    Examples:

ImageRelations    Examples:

InCcGroup    Examples:

IndeterminateAndExponentOfUnivariateMonomial    Examples:

IndeterminateDegrees    Examples:

IndeterminatesOfGradedAlgebraPresentation    Examples:

IndeterminatesOfPolynomial    Examples:

IndexInSL2O    Examples: 1 

InnerAutomorphismGroupQuandle    Examples:

InnerAutomorphismGroupQuandleAsPerm    Examples:

InverseRingHomomorphism    Examples:

IsConnected    Examples: 1 , 2 , 3 

IsHomogeneousQuandle    Examples:

IsLatinQuandle    Examples: 1 

MaximumDegreeForPresentation    Examples:

ModPRingBasisAsPolynomials    Examples:

ModPRingGeneratorDegrees    Examples:

ModPRingNiceBasis    Examples:

ModPRingNiceBasisAsPolynomials    Examples:

Module    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 

NormOfIdeal    Examples:

OuterAction    Examples:

OuterGroup    Examples: 1 , 2 , 3 , 4 

PresentationIdeal    Examples:

PresentationOfGradedStructureConstantAlgebra    Examples: 1 

Pullbacks    Examples:

Pushouts    Examples:

RightMultiplicationGroupOfQuandle    Examples: 1 , 2 , 3 

RightMultiplicationGroupOfQuandleAsPerm    Examples: 1 

SingularGroebnerBasis    Examples:

SingularReducedGroebnerBasis    Examples:

SourceGenerators    Examples:

SourcePolynomialRing    Examples:

SourceRelations    Examples:

StarGraphAttr    Examples:

TermsOfPolynomial    Examples:

UnivariateMonomialsOfMonomial    Examples:

CoefficientsRing    Examples:

ElementsFamily    Examples:

IndexInSL2Z    Examples:

Name    Examples: 1 , 2 , 3 , 4 , 5 , 6 

Order    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 

IsAbelianCategory    Examples:

IsAdditiveCategory    Examples:

IsCategoryName    Examples:

IsCcGroup    Examples:

IsCrystTranslationSubGroup    Examples:

IsGOuterGroup    Examples:

IsGOuterGroupHomomorphism    Examples:

IsGammaSubgroupInSL3Z    Examples:

IsHAPRationalMatrixGroup    Examples:

IsHAPRationalSpecialLinearGroup    Examples:

IsIdealOfQuadraticIntegers    Examples:

IsPeriodic    Examples: 1 , 2 

IsPseudoListWithFunction    Examples:

IsQuadraticNumberField    Examples:

IsRingOfQuadraticIntegers    Examples:

IsStandard2Cocycle    Examples:

IsStandardNCocycle    Examples:

IsCcElement    Examples:

IsGradedAlgebraPresentation    Examples:

IsHAPDerivation    Examples:

IsHAPRingHomomorphism    Examples:

IsHAPRingModIdealObj    Examples:

IsHapCatOneGroup    Examples:

IsHapCatOneGroupMorphism    Examples:

IsHapChainComplex    Examples:

IsHapChainMap    Examples:

IsHapCochainComplex    Examples:

IsHapCochainMap    Examples:

IsHapCommutativeDiagram    Examples:

IsHapConjQuandElt    Examples:

IsHapCrossedModule    Examples:

IsHapCrossedModuleMorphism    Examples:

IsHapCubicalComplex    Examples:

IsHapEquivariantCWComplex    Examples:

IsHapEquivariantChainComplex    Examples:

IsHapEquivariantChainMap    Examples:

IsHapEquivariantNonFreeChainComplex    Examples:

IsHapEquivariantSpectralSequencePage    Examples:

IsHapFilteredChainComplex    Examples:

IsHapFilteredCubicalComplex    Examples:

IsHapFilteredGraph    Examples:

IsHapFilteredPureCubicalComplex    Examples:

IsHapFilteredRegularCWComplex    Examples:

IsHapFilteredSimplicialComplex    Examples:

IsHapFilteredSparseChainComplex    Examples:

IsHapGCocomplex    Examples:

IsHapGComplex    Examples:

IsHapGComplexMap    Examples:

IsHapGraph    Examples:

IsHapOppositeElement    Examples:

IsHapPureCubicalComplex    Examples:

IsHapPureCubicalLink    Examples:

IsHapPurePermutahedralComplex    Examples:

IsHapQuandlePresentation    Examples:

IsHapQuotientElement    Examples:

IsHapRegularCWComplex    Examples:

IsHapRegularCWMap    Examples:

IsHapResolution    Examples:

IsHapSimplicialComplex    Examples:

IsHapSimplicialFreeAbelianGroup    Examples:

IsHapSimplicialGroup    Examples:

IsHapSimplicialGroupMorphism    Examples:

IsHapSimplicialMap    Examples:

IsHapSparseChainComplex    Examples:

IsHapSparseChainMap    Examples:

IsHapSparseMat    Examples:

IsHapTorsionSubcomplex    Examples:

IsPseudoList    Examples:

IsCcElementRep    Examples:

IsGradedAlgebraPresentationRep    Examples:

IsHAPDerivationRep    Examples:

IsHAPIdealRep    Examples:

IsHAPRingHomomorphismIndeterminateMapRep    Examples:

IsHAPRingReductionHomomorphismRep    Examples:

IsHAPRingToSubringHomomorphismRep    Examples:

IsHAPSubringToRingHomomorphismRep    Examples:

IsHAPZeroRingHomomorphismRep    Examples:

IsHapCatOneGroupMorphismRep    Examples:

IsHapCatOneGroupRep    Examples:

IsHapChainComplexRep    Examples:

IsHapChainMapRep    Examples:

IsHapCochainComplexRep    Examples:

IsHapCochainMapRep    Examples:

IsHapCommutativeDiagramRep    Examples:

IsHapConjQuandEltRep    Examples:

IsHapCrossedModuleMorphismRep    Examples:

IsHapCrossedModuleRep    Examples:

IsHapCubicalComplexRep    Examples:

IsHapEquivariantCWComplexRep    Examples:

IsHapEquivariantChainComplexRep    Examples:

IsHapEquivariantChainMapRep    Examples:

IsHapEquivariantNonFreeChainComplexRep    Examples:

IsHapEquivariantSpectralSequencePageRep    Examples:

IsHapFilteredChainComplexRep    Examples:

IsHapFilteredCubicalComplexRep    Examples:

IsHapFilteredGraphRep    Examples:

IsHapFilteredPureCubicalComplexRep    Examples:

IsHapFilteredRegularCWComplexRep    Examples:

IsHapFilteredSimplicialComplexRep    Examples:

IsHapFilteredSparseChainComplexRep    Examples:

IsHapGCocomplexRep    Examples:

IsHapGComplexMapRep    Examples:

IsHapGComplexRep    Examples:

IsHapGraphRep    Examples:

IsHapOppositeElementRep    Examples:

IsHapPureCubicalComplexRep    Examples:

IsHapPureCubicalLinkRep    Examples:

IsHapPurePermutahedralComplexRep    Examples:

IsHapQuandlePresentationRep    Examples:

IsHapQuotientElementRep    Examples:

IsHapRegularCWComplexRep    Examples:

IsHapRegularCWMapRep    Examples:

IsHapResolutionRep    Examples:

IsHapSimplicialComplexRep    Examples:

IsHapSimplicialFreeAbelianGroupRep    Examples:

IsHapSimplicialGroupMorphismRep    Examples:

IsHapSimplicialGroupRep    Examples:

IsHapSimplicialMapRep    Examples:

IsHapSparseChainComplexRep    Examples:

IsHapSparseChainMapRep    Examples:

IsHapSparseMatRep    Examples:

IsHapTorsionSubcomplexRep    Examples:

IsPseudoListRep    Examples:

IdealOfQuadraticIntegers    Examples:

QuadraticNF    Examples:

RingOfQuadraticIntegers    Examples:

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

+    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 

+    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 

+    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 

    Examples:

    Examples:

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

AdditiveInverseMutable    Examples:

AsFpGroup    Examples:

AsList    Examples:

AsSSortedList    Examples:

BarycentricSubdivision    Examples: 1 , 2 

BaseRing    Examples:

Bockstein    Examples: 1 , 2 , 3 

CategoryArrow    Examples:

CategoryObject    Examples:

CoboundaryMatrix    Examples:

CoefficientsOfPoincareSeries    Examples:

CoefficientsRing    Examples:

CohomologicalPeriod    Examples: 1 

CohomologyClass    Examples: 1 , 2 

CompositionRingHomomorphism    Examples:

CompositionRingHomomorphism    Examples:

CompositionRingHomomorphism    Examples:

CompositionRingHomomorphism    Examples:

CompositionRingHomomorphism    Examples:

ConnectedComponentsQuandle    Examples:

ConnectedSum    Examples: 1 , 2 

CoxeterMatrix    Examples: 1 

DefaultFieldOfMatrixGroup    Examples:

DegreeOfRepresentative    Examples:

DerivationImages    Examples:

DerivationRelations    Examples:

DerivationRing    Examples:

Dimensions    Examples:

Enumerator    Examples:

ExcisedPair    Examples:

FilteredRegularCWComplex    Examples: 1 

FundamentalGroupWithPathReps    Examples: 1 , 2 

GModuleAsGOuterGroup    Examples: 1 , 2 , 3 , 4 

GOuterGroupHomomorphism    Examples: 1 , 2 

GOuterGroupHomomorphism    Examples: 1 , 2 

Generators    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 

GeneratorsOfMagmaWithInverses    Examples:

GeneratorsOfMagmaWithInverses    Examples:

GeneratorsOfPresentationIdeal    Examples:

GradedAlgebraPresentation    Examples:

GradedAlgebraPresentationNC    Examples:

HAPDerivationNC    Examples:

HAPPRIME_HilbertSeries    Examples:

HAPRingHomomorphismByIndeterminateMap    Examples:

HAPRingReductionHomomorphism    Examples:

HAPRingReductionHomomorphism    Examples:

HAPRingToSubringHomomorphism    Examples:

HAPSubringToRingHomomorphism    Examples:

HAPSubringToRingHomomorphism    Examples:

HAPZeroRingHomomorphism    Examples:

HAP_MultiplicativeGenerators    Examples:

HomomorphismsImages    Examples:

IdGroup    Examples: 1 , 2 , 3 , 4 , 5 , 6 

IdentityMap    Examples:

ImageOfDerivation    Examples:

ImageOfRingHomomorphism    Examples:

ImageOfRingHomomorphism    Examples:

ImageOfRingHomomorphism    Examples:

ImageOfRingHomomorphism    Examples:

ImageOfRingHomomorphism    Examples:

IndeterminateAndExponentOfUnivariateMonomial    Examples:

IndeterminateDegrees    Examples:

IndeterminatesOfGradedAlgebraPresentation    Examples:

IndeterminatesOfPolynomial    Examples:

IndexInSL2O    Examples: 1 

IndexInSL2Z    Examples:

InnerAutomorphismGroupQuandle    Examples:

InnerAutomorphismGroupQuandleAsPerm    Examples:

Int    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 

InverseMutable    Examples:

InverseMutable    Examples:

InverseMutable    Examples:

InverseRingHomomorphism    Examples:

InverseRingHomomorphism    Examples:

InverseRingHomomorphism    Examples:

InverseRingHomomorphism    Examples:

InverseRingHomomorphism    Examples:

InverseSameMutability    Examples:

IsAssociatedGradedRing    Examples:

IsConnected    Examples: 1 , 2 , 3 

IsHomogeneousQuandle    Examples:

IsLatinQuandle    Examples: 1 

IsMonomial    Examples:

IsOne    Examples:

IsPeriodic    Examples: 1 , 2 

Kernel    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 

KernelOfDerivation    Examples:

MaximumDegreeForPresentation    Examples:

ModPRingBasisAsPolynomials    Examples:

ModPRingGeneratorDegrees    Examples:

ModPRingNiceBasis    Examples:

ModPRingNiceBasisAsPolynomials    Examples:

OneImmutable    Examples:

OneMutable    Examples:

PathComponents    Examples: 1 

PersistentBettiNumbersAlt    Examples: 1 

PreimageOfRingHomomorphism    Examples:

PresentationIdeal    Examples:

PresentationOfGradedStructureConstantAlgebra    Examples: 1 

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

Projection    Examples:

PureComplexMeet    Examples:

PureComplexRandomCell    Examples:

PureComplexSubcomplex    Examples:

Pushout    Examples:

QuadraticIdeal    Examples: 1 

Random    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 

ReduceIdeal    Examples:

ReducedPolynomialRingPresentation    Examples:

ReducedPolynomialRingPresentationMap    Examples:

RefinedColouring    Examples:

Resolution    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 

RightMultiplicationGroupOfQuandle    Examples: 1 , 2 , 3 

RightMultiplicationGroupOfQuandleAsPerm    Examples: 1 

RightTransversal    Examples:

RingOfIntegers    Examples: 1 

SingularGroebnerBasis    Examples:

SingularPolynomialNormalForm    Examples:

SingularReducedGroebnerBasis    Examples:

SingularSetNormalFormIdeal    Examples:

SingularSetNormalFormIdealNC    Examples:

SparseChainComplexOfPair    Examples:

Sq    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 

Standard2Cocycle    Examples:

Standard2Cocycle    Examples:

StandardNCocycle    Examples:

StandardNCocycle    Examples:

StarGraph    Examples:

StarGraphAttr    Examples:

SubspaceBasisRepsByDegree    Examples:

SubspaceDimensionDegree    Examples:

Suspension    Examples: 1 , 2 , 3 , 4 , 5 

TensorProductOp    Examples:

TermsOfPolynomial    Examples:

TrivialGModuleAsGOuterGroup    Examples: 1 , 2 , 3 , 4 

Units    Examples:

Units    Examples:

UnivariateMonomialsOfMonomial    Examples:

VertexLink    Examples:

VertexStar    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

WedgeSum    Examples: 1 

ZeroMutable    Examples:

^    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 

^    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 

^    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

mod    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

+    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 

+    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 

+    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 

-    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

-    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

/    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

    Examples:

    Examples:

    Examples:

    Examples:

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

AbelianInvariants    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 

AbelianInvariants    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 

BarycentricSubdivision    Examples: 1 , 2 

Bockstein    Examples: 1 , 2 , 3 

CanonicalRightCosetElement    Examples:

ClosedSurface    Examples: 1 

CohomologyRing    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 

CohomologyRing    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 

ComplexProjectiveSpace    Examples: 1 

ConnectedSum    Examples: 1 , 2 

ConnectedSum    Examples: 1 , 2 

ConnectedSum    Examples: 1 , 2 

Dimensions    Examples:

DirectProductOp    Examples:

DirectProductOp    Examples:

DirectProductOp    Examples:

DirectProductOp    Examples:

Discriminant    Examples:

Discriminant    Examples:

Embedding    Examples:

GDerivedSubgroup    Examples:

Generators    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 

HAPRingReductionHomomorphism    Examples:

HAPRingReductionHomomorphism    Examples:

HAP_EquivalenceClasses    Examples:

ImageOfRingHomomorphism    Examples:

ImageOfRingHomomorphism    Examples:

IndexNC    Examples:

IndexNC    Examples:

InverseMutable    Examples:

InverseMutable    Examples:

InverseMutable    Examples:

InverseSameMutability    Examples:

IsEmpty    Examples:

IsEmpty    Examples:

IsPrime    Examples: 1 , 2 

IsomorphismFpGroup    Examples: 1 , 2 

Iterator    Examples:

KernelOfDerivation    Examples:

ListOp    Examples:

ListOp    Examples:

LowerGCentralSeries    Examples:

NaturalHomomorphism    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 

Norm    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 

Norm    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 

OneImmutable    Examples:

OneImmutable    Examples:

Order    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 

Order    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 

PersistentBettiNumbersAlt    Examples: 1 

PersistentBettiNumbersAlt    Examples: 1 

PersistentBettiNumbersAlt    Examples: 1 

PersistentBettiNumbersAlt    Examples: 1 

PersistentBettiNumbersAlt    Examples: 1 

PersistentHomology    Examples: 1 , 2 

PersistentHomology    Examples: 1 , 2 

Position    Examples: 1 , 2 

Position    Examples: 1 , 2 

Position    Examples: 1 , 2 

PositionCanonical    Examples:

PreimageOfRingHomomorphism    Examples:

Projection    Examples:

PureComplexMeet    Examples:

PureComplexRandomCell    Examples:

PureComplexSubcomplex    Examples:

QuadraticIdeal    Examples: 1 

Range    Examples: 1 , 2 

RankMatrixDestructive    Examples:

ReduceIdeal    Examples:

ReduceIdeal    Examples:

ReducedPolynomialRingPresentation    Examples:

ReducedPolynomialRingPresentation    Examples:

ReducedPolynomialRingPresentationMap    Examples:

ReducedPolynomialRingPresentationMap    Examples:

ReducedPolynomialRingPresentationMap    Examples:

RefinedColouring    Examples:

RightTransversal    Examples:

RightTransversal    Examples:

RightTransversal    Examples:

RightTransversal    Examples:

RightTransversal    Examples:

RightTransversal_alt    Examples:

SingularPolynomialNormalForm    Examples:

SparseChainComplexOfPair    Examples:

Sphere    Examples: 1 

SubspaceBasisRepsByDegree    Examples:

SubspaceDimensionDegree    Examples:

Suspension    Examples: 1 , 2 , 3 , 4 , 5 

TensorProductOp    Examples:

TensorProductOp    Examples:

TensorProductOp    Examples:

Trace    Examples:

Units    Examples:

WedgeSum    Examples: 1 

WedgeSum    Examples: 1 

WedgeSum    Examples: 1 

[]    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 

[]    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 

[]    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 

^    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 

^    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 

^    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 

^    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

mod    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 

InfoHAPprime    Examples:

ASY_PATH    Examples:

AutomorphismGroupAsCrossedModule    Examples:

BROWSER_PATH    Examples:

CATONEGROUP_DATA_PERM    Examples:

CATONEGROUP_DATA_SIZE    Examples:

Cedric_PlanarDiagram    Examples:

ChildKill    Examples:

DISPLAY_PATH    Examples:

DOT_PATH    Examples:

FilteredSimplicialComplexToFilteredCWComplex    Examples:

GradedAlgebraPresentationType    Examples:

HAPTEMPORARYFUNCTION    Examples:

HAP_Knots    Examples:

HAP_ROOT    Examples:

HapCatOneGroup    Examples:

HapCatOneGroupFamily    Examples:

HapCatOneGroupMorphism    Examples:

HapCatOneGroupMorphismFamily    Examples:

HapChainComplex    Examples:

HapChainComplexFamily    Examples:

HapChainMap    Examples:

HapChainMapFamily    Examples:

HapCochainComplex    Examples:

HapCochainComplexFamily    Examples:

HapCochainMap    Examples:

HapCochainMapFamily    Examples:

HapCommutativeDiagram    Examples:

HapCommutativeDiagramFamily    Examples:

HapCrossedModule    Examples:

HapCrossedModuleFamily    Examples:

HapCrossedModuleMorphism    Examples:

HapCrossedModuleMorphismFamily    Examples:

HapCubicalComplex    Examples:

HapCubicalComplexFamily    Examples:

HapEquivariantCWComplex    Examples:

HapEquivariantCWComplexFamily    Examples:

HapEquivariantChainMap    Examples:

HapEquivariantChainMapFamily    Examples:

HapFPGModule    Examples:

HapFPGModuleHomomorphism    Examples:

HapFilteredChainComplex    Examples:

HapFilteredChainComplexFamily    Examples:

HapFilteredCubicalComplex    Examples:

HapFilteredCubicalComplexFamily    Examples:

HapFilteredGraph    Examples:

HapFilteredGraphFamily    Examples:

HapFilteredPureCubicalComplex    Examples:

HapFilteredPureCubicalComplexFamily    Examples:

HapFilteredRegularCWComplex    Examples:

HapFilteredRegularCWComplexFamily    Examples:

HapFilteredSimplicialComplex    Examples:

HapFilteredSimplicialComplexFamily    Examples:

HapFilteredSparseChainComplex    Examples:

HapFilteredSparseChainComplexFamily    Examples:

HapGChainComplex    Examples:

HapGCocomplex    Examples:

HapGCocomplexFamily    Examples:

HapGComplex    Examples:

HapGComplexFamily    Examples:

HapGlobalDeclarationsAreAlreadyLoaded    Examples:

HapGraph    Examples:

HapGraphFamily    Examples:

HapNonFreeResolution    Examples:

HapOppositeElement    Examples:

HapOppositeElementFamily    Examples:

HapPureCubicalComplex    Examples:

HapPureCubicalComplexFamily    Examples:

HapPureCubicalLink    Examples:

HapPureCubicalLinkFamily    Examples:

HapPurePermutahedralComplex    Examples:

HapPurePermutahedralComplexFamily    Examples:

HapQuotientElement    Examples:

HapQuotientElementFamily    Examples:

HapRegularCWComplex    Examples:

HapRegularCWComplexFamily    Examples:

HapRegularCWMap    Examples:

HapRegularCWMapFamily    Examples:

HapResolution    Examples:

HapResolutionFamily    Examples:

HapSimplicialComplex    Examples:

HapSimplicialComplexFamily    Examples:

HapSimplicialGroup    Examples:

HapSimplicialGroupFamily    Examples:

HapSimplicialGroupMorphism    Examples:

HapSimplicialGroupMorphismFamily    Examples:

HapSimplicialMap    Examples:

HapSimplicialMapFamily    Examples:

HapSparseChainComplex    Examples:

HapSparseChainComplexFamily    Examples:

HapSparseChainMap    Examples:

HapSparseChainMapFamily    Examples:

HapSparseMat    Examples:

HapSparseMatFamily    Examples:

HomomorphismOfDirectProduct    Examples:

IDQUASICATONEGROUP_DATA    Examples:

IsHapChain    Examples:

IsHapCochain    Examples:

IsHapComplex    Examples:

IsHapFPGModule    Examples:

IsHapFPGModuleHomomorphism    Examples:

IsHapGChainComplex    Examples:

IsHapMap    Examples:

IsHapNonFreeResolution    Examples:

NEATO_PATH    Examples:

NerveOfCover    Examples:

POLYMAKE_PATH    Examples:

PseudoList    Examples:

PseudoListFamily    Examples:

QUASICATONEGROUP_DATA_NOT    Examples:

QUASICATONEGROUP_DATA_SIZE    Examples:

ReadBioData    Examples:

SMALLQUASICATONEGROUP_DATA    Examples:

CATONEGROUP_DATA    Examples:

COMPILED    Examples:

Cedric_XYXYConnQuan    Examples:

Cedric_XYXYQuandles    Examples:

CommutingProbability    Examples:

GroupIsomorphismRepresentatives    Examples:

HAPAAA    Examples:

HAPBARCODE    Examples:

HAPDerivationType    Examples:

HAPPRIME_LastLHSBicomplexSize    Examples:

HAPPRIME_ShuffleRandomSource    Examples:

HAPRIGXXX    Examples:

HAP_GCOMPLEX_LIST    Examples:

HAP_GCOMPLEX_SETUP    Examples:

HAP_MOVES_DIM_2    Examples:

HAP_MOVES_DIM_3    Examples:

HAP_PERMMOVES_DIM_2    Examples:

HAP_PERMMOVES_DIM_3    Examples:

HAP_PoincareCubeManifoldEdgeDegrees    Examples:

HAP_Test    Examples:

HAP_XYXYXYXY    Examples:

HAPchildFunctionToggle    Examples:

HAPchildToggle    Examples:

HAPchildren    Examples:

HapConjQuandElt    Examples:

HapConjQuandEltFamily    Examples:

HapConstantPolRing    Examples:

HapEquivariantChainComplex    Examples:

HapEquivariantChainComplexFamily    Examples:

HapEquivariantNonFreeChainComplex    Examples:

HapEquivariantNonFreeChainComplexFamily    Examples:

HapEquivariantSpectralSequencePage    Examples:

HapEquivariantSpectralSequencePageFamily    Examples:

HapGComplexMap    Examples:

HapGComplexMapFamily    Examples:

HapQuandlePresentation    Examples:

HapQuandlePresentationFamily    Examples:

HapRightTransversalSL2ZSubgroup    Examples:

HapSL2ZConjugatedSubgroup    Examples:

HapSL2ZSubgroup    Examples:

HapSimplicialFreeAbelianGroup    Examples:

HapSimplicialFreeAbelianGroupFamily    Examples:

HapTorsionSubcomplex    Examples:

HapTorsionSubcomplexFamily    Examples:

IntersectionForm    Examples: 1 , 2 

IsHapRightTransversalSL2ZSubgroup    Examples:

IsHapSL2ConjugatedSubgroup    Examples:

IsHapSL2OSubgroup    Examples:

IsHapSL2Subgroup    Examples:

IsHapSL2ZConjugatedSubgroup    Examples:

IsHapSL2ZSubgroup    Examples:

RefinedColouring_gc    Examples:

RefinedColouring_group    Examples:

RegularCWAssociahedronWithDiscreteVectorField    Examples:

RegularCWClosedSurface    Examples:

RegularCWComplexWithAttachedRelatorCells    Examples: 1 

RegularCWComplex_DisjointUnion    Examples:

RegularCWComplex_WedgeSum    Examples:

RegularCWDiscreteSpace    Examples: 1 

RegularCWSphere    Examples: 1 

SimplicialComplexConnectedSum    Examples:

SphericalKnotComplementWithBoundary    Examples:

StemGroups    Examples:

cat    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 

cnt    Examples:

hap_cr    Examples:

+

2CoreducedChainComplex    Examples:

AbelianGOuterGroupToCatOneGroup    Examples:

AbelianInvariantsToTorsionCoefficients    Examples:

AcyclicSubcomplexOfPureCubicalComplex    Examples: 1 

AddFirst    Examples:

AdjointGroupOfQuandle    Examples: 1 

AlgebraicReduction_alt    Examples:

AppendFreeWord    Examples:

ArcDiagramToTubularSurface    Examples:

ArcPresentation    Examples: 1 , 2 , 3 , 4 

ArcPresentationToKnottedOneComplex    Examples:

AreIsoclinic    Examples:

ArrayIterateBreak    Examples:

ArrayValueKD    Examples:

AsWordInSL2Z    Examples:

AutomorphismGroupQuandleAsPerm_nonconnected    Examples:

AverageInnerProduct    Examples:

BarCodeOfFilteredPureCubicalComplex    Examples:

BarCodeOfSymmetricMatrix    Examples:

BarComplexOfMonoid    Examples: 1 

BarycentricallySimplifiedComplex    Examples: 1 

BarycentricallySubdivideCell    Examples:

BettinumbersOfPureCubicalComplex_dim_2    Examples:

BocksteinHomology    Examples:

BogomolovMultiplier_viaTensorSquare    Examples:

BoundariesOfFilteredChainComplex    Examples:

BoundaryOfPureComplex    Examples: 1 

BoundaryOfPureRegularCWComplex    Examples: 1 

BoundaryOfRegularCWCell    Examples:

BoundaryPairOfPureRegularCWComplex    Examples:

BoundingPureComplex    Examples:

CR_ChainMapFromCocycle    Examples:

CR_CocyclesAndCoboundaries    Examples:

CR_IntegralClassToCocycle    Examples:

CR_IntegralCocycleToClass    Examples:

CR_IntegralCohomology    Examples:

CR_IntegralCycleToClass    Examples:

CWMap2ChainMap    Examples:

CWSubcomplexToRegularCWMap    Examples: 1 

CanonicalRightCountableCosetElement    Examples:

CatOneGroupByCrossedModule    Examples:

CatOneGroupsByGroup    Examples:

CcElement    Examples:

Cedric_CheckThirdAxiomRow    Examples:

Cedric_ConjugateQuandleElement    Examples:

Cedric_FromAutGeReToAutQe    Examples:

Cedric_IsHomomorphism    Examples:

Cedric_Permute    Examples:

Cedric_Quandle1    Examples:

Cedric_Quandle2    Examples:

Cedric_Quandle3    Examples:

Cedric_Quandle4    Examples:

Cedric_Quandle5    Examples:

Cedric_Quandle6    Examples:

CellComplexBoundaryCheck    Examples:

ChainComplexEquivalenceOfRegularCWComplex    Examples: 1 

ChainComplexHomeomorphismEquivalenceOfRegularCWComplex    Examples:

ChainComplexOfCubicalComplex    Examples:

ChainComplexOfCubicalPair    Examples:

ChainComplexOfRegularCWComplexWithVectorField    Examples:

ChainComplexOfSimplicialComplex    Examples:

ChainComplexOfSimplicialPair    Examples:

ChainComplexOfUniversalCover    Examples: 1 , 2 , 3 , 4 

ChainComplexToSparseChainComplex    Examples:

ChainComplexWithChainHomotopy    Examples:

ChainMapOfCubicalPairs    Examples:

ChainMapOfRegularCWMap    Examples:

ChevalleyEilenbergComplexOfModule    Examples:

ChildRestart    Examples:

ClosureCWCell    Examples:

CoClass    Examples:

CocriticalCellsOfRegularCWComplex    Examples:

CocyclicHadamardMatrices    Examples: 1 

CocyclicMatrices    Examples:

CohomologicalData    Examples: 1 

CohomologyHomomorphism    Examples: 1 , 2 

CohomologyHomomorphismOfRepresentation    Examples:

CohomologyModule_AsAutModule    Examples:

CohomologyModule_Gmap    Examples:

CohomologyRingOfSimplicialComplex    Examples:

CohomologySimplicialFreeAbelianGroup    Examples:

CombinationDisjointSets    Examples:

CommonEndomorphisms    Examples:

ComplementOfPureComplex    Examples: 1 

ComplementaryBasis    Examples:

ComposeCWMaps    Examples:

CompositionOfFpGModuleHomomorphisms    Examples:

CompositionSeriesOfFpGModule    Examples:

ConcentricallyFilteredPureCubicalComplex    Examples: 1 

CongruenceSubgroup    Examples: 1 , 2 

ConjugateSL2ZGroup    Examples:

ConnectingCohomologyHomomorphism    Examples: 1 , 2 

ContractArray    Examples:

ContractCubicalComplex_dim2    Examples:

ContractCubicalComplex_dim3    Examples:

ContractMatrix    Examples:

ContractPermArray    Examples:

ContractPermMatrix    Examples:

ContractPureComplex    Examples:

ContractSimplicialComplex    Examples:

ContractSimplicialComplex_alt    Examples:

ContractedFilteredPureCubicalComplex    Examples: 1 

ContractedFilteredRegularCWComplex    Examples:

ContractedRegularCWComplex    Examples:

ContractibleSL2ZComplex    Examples:

ContractibleSL2ZComplex_alt    Examples:

ContractibleSubArray    Examples:

ContractibleSubMatrix    Examples:

ContractibleSubcomplexOfPureCubicalComplex    Examples: 1 

ConvertTorsionComplexToGcomplex    Examples:

CosetsQuandle    Examples:

CountingCellsOfBaryCentricSubdivision    Examples:

CountingNumberOfCellsInBaryCentricSubdivision    Examples:

CoxeterComplex_alt    Examples: 1 

CoxeterDiagramMatCoxeterGroup    Examples:

CoxeterWythoffComplex    Examples:

CreateCoxeterMatrix    Examples: 1 

CriticalBoundaryCells    Examples: 1 

CropPureComplex    Examples:

CrossedInvariant    Examples:

CrossedModuleByAutomorphismGroup    Examples:

CrossedModuleByCatOneGroup    Examples:

CrossedModuleByNormalSubgroup    Examples: 1 

CrystCubicalTiling    Examples:

CrystFinitePartOfMatrix    Examples:

CrystGFullBasis    Examples: 1 , 2 

CrystGcomplex    Examples: 1 , 2 

CrystMatrix    Examples:

CrystTranslationMatrixToVector    Examples:

CrystallographicComplex    Examples:

CubicalToPermutahedralArray    Examples:

CupProductMatrix    Examples:

CupProductOfRegularCWComplex    Examples: 1 

CupProductOfRegularCWComplex_alt    Examples: 1 

CuspidalCohomologyHomomorphism    Examples:

CyclesOfFilteredChainComplex    Examples:

DavisComplex    Examples: 1 , 2 , 3 , 4 

DeformationRetract    Examples:

DensityMat    Examples:

DerivedGroupOfQuandle    Examples: 1 

DiagonalChainMap    Examples:

DijkgraafWittenInvariant    Examples: 1 

DirectProductOfGroupHomomorphisms    Examples:

DirectProductOfRegularCWComplexes    Examples:

DirectProductOfRegularCWComplexesLazy    Examples:

DirectProductOfSimplicialComplexes    Examples:

DisplayCSVknotFile    Examples:

DisplayVectorField    Examples:

E1CohomologyPage    Examples:

E1HomologyPage    Examples:

EilenbergMacLaneSimplicialFreeAbelianGroup    Examples:

ElementsLazy    Examples:

EquivariantCWComplexToRegularCWComplex    Examples: 1 , 2 , 3 , 4 

EquivariantCWComplexToRegularCWMap    Examples: 1 , 2 , 3 

EquivariantCWComplexToResolution    Examples:

ExcisedPureCubicalPair_dim_2    Examples:

ExtractTorsionSubcomplex    Examples:

FactorizationNParts    Examples:

FilteredChainComplexToFilteredSparseChainComplex    Examples:

FilteredCubicalComplexToFilteredRegularCWComplex    Examples: 1 

FilteredPureCubicalComplexToCubicalComplex    Examples: 1 

FiltrationTermOfGraph    Examples:

FiltrationTermOfPureCubicalComplex    Examples:

FiltrationTermOfRegularCWComplex    Examples:

FiltrationTerms    Examples: 1 

FirstHomologyCoveringCokernels    Examples: 1 , 2 

FirstHomologySimplicialTwoComplex    Examples:

FourthHomotopyGroupOfDoubleSuspensionB    Examples:

Fp2PcpAbelianGroupHomomorphism    Examples:

FpGModuleSection    Examples:

FreeZGResolution    Examples:

FundamentalGroupOfRegularCWComplex    Examples: 1 

FundamentalGroupOfRegularCWMap    Examples:

FundamentalGroupSimplicialTwoComplex    Examples:

FundamentalMultiplesOfStiefelWhitneyClasses    Examples:

GChainComplex    Examples: 1 

GModuleAsCatOneGroup    Examples:

GammaSubgroupInSL3Z    Examples:

GaussCodeOfPureCubicalKnot    Examples: 1 , 2 , 3 , 4 

GetTorsionPowerSubcomplex    Examples:

GetTorsionSubcomplex    Examples:

GraphOfRegularCWComplex    Examples:

GraphOfResolutionsTest    Examples:

GraphOfResolutionsToGroups    Examples:

GroupHomomorphismToMatrix    Examples:

HAPCocontractRegularCWComplex    Examples:

HAPContractFilteredRegularCWComplex    Examples:

HAPContractRegularCWComplex    Examples:

HAPContractRegularCWComplex_Alt    Examples:

HAPPRIME_Algebra2Polynomial    Examples:

HAPPRIME_CohomologyRingWithoutResolution    Examples:

HAPPRIME_CombineIndeterminateMaps    Examples:

HAPPRIME_GradedAlgebraPresentationAvoidingIndeterminates    Examples:

HAPPRIME_LHSSpectralSequence    Examples:

HAPPRIME_MakeEliminationOrdering    Examples:

HAPPRIME_MapPolynomialIndeterminates    Examples:

HAPPRIME_Polynomial2Algebra    Examples:

HAPPRIME_RingHomomorphismsAreComposable    Examples:

HAPPRIME_SModule    Examples:

HAPPRIME_SingularGroebnerBasis    Examples:

HAPPRIME_SingularReducedGroebnerBasis    Examples:

HAPPRIME_SwitchGradedAlgebraRing    Examples:

HAPPRIME_SwitchPolynomialIndeterminates    Examples:

HAPPRIME_VersionWithSVN    Examples:

HAPRegularCWComplex    Examples:

HAPRegularCWPolytope    Examples:

HAPRemoveCellFromRegularCWComplex    Examples:

HAPRemoveVectorField    Examples:

HAPRingModIdeal    Examples:

HAPRingModIdealObj    Examples:

HAPTietzeReduction_Inf    Examples:

HAPTietzeReduction_OneLevel    Examples:

HAPTietzeReduction_OneStep    Examples:

HAP_4x4MatTo2x2Mat    Examples:

HAP_AddGenerator    Examples:

HAP_AllHomomorphisms    Examples:

HAP_AppendTo    Examples:

HAP_AssociahedronBoundaries    Examples:

HAP_AssociahedronCells    Examples:

HAP_BaryCentricSubdivisionGComplex    Examples:

HAP_BaryCentricSubdivisionRegularCWComplex    Examples:

HAP_Binlisttoint    Examples:

HAP_ChainComplexToEquivariantChainComplex    Examples:

HAP_CocyclesAndCoboundaries    Examples:

HAP_CongruenceSubgroupGamma0    Examples: 1 

HAP_CongruenceSubgroupGamma0Ideal    Examples:

HAP_ConjugatedCongruenceSubgroup    Examples:

HAP_ConjugatedCongruenceSubgroupGamma0    Examples:

HAP_CriticalCellsDirected    Examples:

HAP_CupProductOfPresentation    Examples:

HAP_CupProductOfSimplicialComplex    Examples:

HAP_DisplayPlanarTree    Examples:

HAP_DisplayVectorField    Examples:

HAP_ElementsSL2Zfn    Examples:

HAP_FunctorialModPCohomologyRing    Examples:

HAP_GenericSL2OSubgroup    Examples:

HAP_GenericSL2ZConjugatedSubgroup    Examples:

HAP_GenericSL2ZSubgroup    Examples:

HAP_HomToIntModP_ChainComplex    Examples:

HAP_HomToIntModP_ChainMap    Examples:

HAP_HomToIntModP_CochainComplex    Examples:

HAP_HomToIntModP_CochainMap    Examples:

HAP_HomeoLinkingForm    Examples:

HAP_Hurewicz1Cycles    Examples:

HAP_IntegralClassToCocycle    Examples:

HAP_IntegralCocycleToClass    Examples:

HAP_IntegralCohomology    Examples:

HAP_KK_AddCell    Examples:

HAP_KnotGroupInv    Examples:

HAP_MyIsBieberbachFpGroup    Examples:

HAP_MyIsFiniteFpGroup    Examples:

HAP_MyIsInfiniteFpGroup    Examples:

HAP_PHI    Examples:

HAP_PermBinlisttoint    Examples:

HAP_PlanarBinaryTrees    Examples:

HAP_PlanarTreeGraft    Examples:

HAP_PlanarTreeJoin    Examples:

HAP_PlanarTreeLeaves    Examples:

HAP_PlanarTreeRemovableEdge    Examples:

HAP_PlanarTreeRemoveEdge    Examples:

HAP_PrimePartModified    Examples:

HAP_PrincipalCongruenceSubgroup    Examples:

HAP_PrincipalCongruenceSubgroupIdeal    Examples:

HAP_PrintTo    Examples:

HAP_PureComplexSubcomplex    Examples:

HAP_PureCubicalPairToCWMap    Examples:

HAP_ResolutionAbelianGroupFromInvariants    Examples:

HAP_RightTransversalSL2ZSubgroups    Examples:

HAP_SL2OSubgroupTree_slow    Examples:

HAP_SL2SubgroupTree    Examples:

HAP_SL2TreeDisplay    Examples:

HAP_SL2ZSubgroupTree_fast    Examples:

HAP_SL2ZSubgroupTree_slow    Examples:

HAP_Sequence2Boundaries    Examples:

HAP_SimplicialPairToCWMap    Examples:

HAP_SimplicialProjectivePlane    Examples:

HAP_SimplicialTorus    Examples:

HAP_SimplifiedGaussCode    Examples:

HAP_StiefelWhitney    Examples:

HAP_SylowSubgroups    Examples:

HAP_Tensor    Examples:

HAP_TransversalCongruenceSubgroups    Examples:

HAP_TransversalCongruenceSubgroupsIdeal    Examples:

HAP_TransversalCongruenceSubgroupsIdeal_alt    Examples:

HAP_TransversalGamma0SubgroupsIdeal    Examples:

HAP_Triangulation    Examples:

HAP_TzPair    Examples:

HAP_WedgeSumOfSimplicialComplexes    Examples:

HAP_bockstein    Examples:

HAP_chain_bockstein    Examples:

HAP_coho_isoms    Examples:

HAP_nxnMatTo2nx2nMat    Examples:

HadamardGraph    Examples:

HapExample    Examples:

HapFile    Examples: 1 , 2 , 3 , 4 

HasTrivialPostnikovInvariant    Examples:

HeckeOperator    Examples:

HeckeOperatorWeight2    Examples:

HenonOrbit    Examples: 1 

HomToGModule_hom    Examples:

HomToInt_ChainComplex    Examples:

HomToInt_ChainMap    Examples:

HomToInt_CochainComplex    Examples:

HomToModPModule    Examples: 1 

HomogeneousPolynomials    Examples:

HomogeneousPolynomials_Bianchi    Examples:

HomologicalGroupDecomposition    Examples: 1 

HomologyOfPureCubicalComplex    Examples:

HomologyPbs    Examples:

HomologySimplicialFreeAbelianGroup    Examples:

HomomorphismAsMatrix    Examples:

HomotopyCatOneGroup    Examples:

HomotopyCrossedModule    Examples:

HomotopyEquivalentLargerSubArray    Examples:

HomotopyEquivalentLargerSubArray3D    Examples:

HomotopyEquivalentLargerSubMatrix    Examples:

HomotopyEquivalentLargerSubPermArray    Examples:

HomotopyEquivalentLargerSubPermArray3D    Examples:

HomotopyEquivalentLargerSubPermMatrix    Examples:

HomotopyEquivalentMaximalPureSubcomplex    Examples:

HomotopyEquivalentMinimalPureSubcomplex    Examples:

HomotopyEquivalentSmallerSubArray    Examples:

HomotopyEquivalentSmallerSubArray3D    Examples:

HomotopyEquivalentSmallerSubMatrix    Examples:

HomotopyEquivalentSmallerSubPermArray    Examples:

HomotopyEquivalentSmallerSubPermArray3D    Examples:

HomotopyEquivalentSmallerSubPermMatrix    Examples:

HomotopyLowerCentralSeries    Examples:

HomotopyLowerCentralSeriesOfCrossedModule    Examples:

HomotopyTruncation    Examples:

HopfSatohSurface    Examples: 1 , 2 

HybridSubdivision    Examples:

IdCatOneGroup    Examples: 1 

IdCrossedModule    Examples:

IdQuasiCatOneGroup    Examples:

IdQuasiCrossedModule    Examples:

IdentifyKnot    Examples: 1 

IdentityAmongRelators    Examples: 1 , 2 , 3 

ImageOfGOuterGroupHomomorphism    Examples: 1 , 2 

ImageOfMap    Examples:

InducedSteenrodHomomorphisms    Examples:

IntegerSimplicialComplex    Examples: 1 

IntegralCellularHomology    Examples:

IntegralCohomology    Examples:

IntegralCohomologyOfCochainComplex    Examples:

IntegralHomology    Examples: 1 

IntegralHomologyOfChainComplex    Examples:

IntersectionCWSubcomplex    Examples:

IsClosedManifold    Examples: 1 

IsContractibleCube_higherdims    Examples:

IsCrystSameOrbit    Examples:

IsCrystSufficientLattice    Examples:

IsHadamardMatrix    Examples:

IsIntList    Examples:

IsIsomorphismOfAbelianFpGroups    Examples: 1 

IsMetricMatrix    Examples:

IsPeriodicSpaceGroup    Examples: 1 

IsPureComplex    Examples:

IsPureRegularCWComplex    Examples:

IsRigid    Examples: 1 

IsRigidOnRight    Examples:

IsSphericalCoxeterGroup    Examples:

IsoclinismClasses    Examples: 1 , 2 

IsomorphismCatOneGroups    Examples: 1 

IsomorphismCrossedModules    Examples:

KernelOfGOuterGroupHomomorphism    Examples: 1 , 2 

KernelOfMap    Examples:

KernelWG    Examples:

KinkArc2Presentation    Examples:

KnotComplement    Examples: 1 , 2 , 3 

KnotComplementWithBoundary    Examples: 1 , 2 , 3 

LazyList    Examples:

LefschetzNumberOfChainMap    Examples:

Lfunction    Examples:

LiftColouredSurface    Examples:

LiftedRegularCWMap    Examples:

LinearHomomorphismsZZPersistenceMat    Examples:

LinkingForm    Examples: 1 

LinkingFormHomeomorphismInvariant    Examples: 1 

LinkingFormHomotopyInvariant    Examples: 1 

ListsOfCellsToRegularCWComplex    Examples:

LowDimensionalCupProduct    Examples: 1 

MakeHAPprimeDoc    Examples:

ManifoldType    Examples: 1 

Mapper    Examples: 1 

Mapper_alt    Examples:

MatrixSize    Examples:

MaximalSimplicesOfSimplicialComplex    Examples: 1 

MaximalSphericalCoxeterSubgroupsFromAbove    Examples:

MinimizeRingRelations    Examples:

Mod2SteenrodAlgebra    Examples: 1 

ModPCohomologyRing_alt    Examples:

ModPCohomologyRing_part_1    Examples:

ModPCohomologyRing_part_2    Examples:

ModPRingGeneratorsAlt    Examples:

ModPSteenrodAlgebra    Examples: 1 , 2 

ModularCohomology    Examples:

ModularEquivariantChainMap    Examples:

ModularHomology    Examples:

Nil3TensorSquare    Examples:

NonFreeResolutionFiniteSubgroup    Examples:

NonManifoldVertices    Examples:

NonRegularCWBoundary    Examples:

NonabelianSymmetricKernel_alt    Examples: 1 

NonabelianSymmetricSquare_inf    Examples:

NonabelianTensorProduct_Inf    Examples:

NonabelianTensorProduct_alt    Examples:

NonabelianTensorSquareAsCatOneGroup    Examples:

NonabelianTensorSquareAsCrossedModule    Examples:

NonabelianTensorSquare_inf    Examples:

NoncrossingPartitionsLatticeDisplay    Examples: 1 

NullspaceSparseMatDestructive    Examples:

NumberConnectedQuandles    Examples:

NumberGeneratorsOfGroupHomology    Examples:

NumberOfCrossingsInArc2Presentation    Examples:

NumberOfHomomorphisms_connected    Examples:

NumberOfHomomorphisms_groups    Examples:

NumberOfPrimeKnots    Examples: 1 , 2 

NumberSmallCatOneGroups    Examples:

NumberSmallCrossedModules    Examples:

NumberSmallQuasiCatOneGroups    Examples:

NumberSmallQuasiCrossedModules    Examples:

OppositeGroup    Examples:

OrthogonalizeBasisByAverageInnerProduct    Examples:

PCentre    Examples:

PSubgroupGChainComplex    Examples:

PSubgroupSimplicialComplex    Examples:

PUpperCentralSeries    Examples:

PartialIsoclinismClasses    Examples: 1 

PartsOfQuadraticInteger    Examples:

PathComponentOfPureComplex    Examples: 1 

PathComponentsCWSubcomplex    Examples:

PathComponentsOfSimplicialComplex_alt    Examples:

PathObjectForChainComplex    Examples: 1 

PermutahedralComplexToRegularCWComplex    Examples: 1 

PermutahedralToCubicalArray    Examples:

PersistentBettiNumbersViaContractions    Examples:

PersistentHomologyOfCrossedModule    Examples:

PersistentHomologyOfFilteredPureCubicalComplex_alt    Examples:

PersistentHomologyOfFilteredSparseChainComplex    Examples: 1 , 2 

PersistentHomologyOfPureCubicalComplex_Alt    Examples:

PersistentHomologyOfQuotientGroupSeries_Int    Examples:

PiZeroOfRegularCWComplex    Examples:

PoincareBipyramidCWComplex    Examples: 1 

PoincareCubeCWComplex    Examples: 1 

PoincareCubeCWComplexNS    Examples: 1 

PoincareDodecahedronCWComplex    Examples: 1 , 2 

PoincareOctahedronCWComplex    Examples: 1 

PoincarePrismCWComplex    Examples: 1 

PoincareSeriesApproximation    Examples:

PoincareSeries_alt    Examples:

PolymakeFaceLattice    Examples:

PolytopalRepresentationComplex    Examples:

PrankAlt    Examples:

PresentationOfResolution_alt    Examples:

PrimePartDerivedFunctorHomomorphism    Examples:

PrimePartDerivedFunctorViaSubgroupChain    Examples:

PrimePartDerivedTwistedFunctor    Examples:

PrintAlgebraWordAsPolynomial    Examples:

PrintTorsionSubcomplex    Examples:

PureComplex    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 

PureCubicalComplexToCubicalComplex    Examples: 1 , 2 

PureCubicalLink    Examples: 1 , 2 

PushoutOfFpGroups    Examples:

QuadraticCharacter    Examples:

QuadraticNumberField    Examples: 1 

QuandleIsomorphismRepresentatives    Examples:

QuotientByTorsionSubcomplex    Examples:

QuotientChainMap    Examples:

QuotientGroup    Examples:

QuotientQuasiIsomorph    Examples:

RadicalSeriesOfResolution    Examples:

RandomArc2Presentation    Examples:

RandomCellOfPureComplex    Examples:

ReadLinkImageAsGaussCode    Examples: 1 

ReadMatrixAsPureCubicalComplex    Examples:

ReduceGenerators    Examples:

ReduceGenerators_alt    Examples:

ReflectedCubicalKnot    Examples: 1 , 2 , 3 , 4 

RegularCWAssociahedron    Examples:

RegularCWComplexComplement    Examples: 1 

RegularCWComplexWithRemovedCell    Examples: 1 

RegularCWComplex_AttachCellDestructive    Examples: 1 

RegularCWCube    Examples:

RegularCWMapToCWSubcomplex    Examples:

RegularCWOrbitPolytope    Examples:

RegularCWPermutahedron    Examples:

RegularCWPolygon    Examples:

RegularCWSimplex    Examples:

RelativeCentralQuotientSpaceGroup    Examples:

RelativeGroupHomology    Examples:

RelativeRightTransversal    Examples:

RemoveStar    Examples:

ResolutionAbelianGroup_alt    Examples:

ResolutionAbelianPcpGroup    Examples:

ResolutionAffineCrystGroup    Examples:

ResolutionBoundaryOfWordOnRight    Examples:

ResolutionDirectProductLazy    Examples:

ResolutionFiniteCyclicGroup    Examples:

ResolutionGL2QuadraticIntegers    Examples:

ResolutionGL3QuadraticIntegers    Examples:

ResolutionGenericGroup    Examples:

ResolutionInfiniteCyclicGroup    Examples:

ResolutionPGL2QuadraticIntegers    Examples:

ResolutionPGL3QuadraticIntegers    Examples:

ResolutionPSL2QuadraticIntegers    Examples: 1 

ResolutionPrimePowerGroupSparse    Examples:

ResolutionSL2QuadraticIntegers    Examples: 1 

ResolutionSL2ZConjugated    Examples:

ResolutionSL2Z_alt    Examples:

ResolutionSpaceGroup    Examples: 1 

ResolutionToEquivariantCWComplex    Examples:

ResolutionToResolutionOfFpGroup    Examples: 1 

SL2QuadraticIntegers    Examples: 1 

SL2ZResolution    Examples:

SL2ZResolution_alt    Examples:

SL2ZTree    Examples:

SL2ZmElementsDecomposition    Examples:

SequentialRegularCWComplexComplement    Examples:

SignatureOfSymmetricMatrix    Examples: 1 

SignedPermutationGroup    Examples: 1 

SimplicesToSimplicialComplex    Examples: 1 , 2 , 3 , 4 

SimplicialComplexToRegularCWComplex_alt    Examples:

SimplicialK3Surface    Examples: 1 

SimplicialNerveOfFilteredGraph    Examples: 1 , 2 

SimplicialNerveOfTwoComplex    Examples:

SimplifiedQuandlePresentation    Examples:

SimplifiedRegularCWComplex    Examples: 1 

SimplifiedSparseChainComplex    Examples:

SmallCatOneGroup    Examples: 1 

SmallCrossedModule    Examples:

SmallQuasiCatOneGroup    Examples:

SmallQuasiCrossedModule    Examples:

SmoothedFpGroup    Examples:

SparseChainComplexOfCubicalComplex    Examples:

SparseChainComplexOfCubicalPair    Examples:

SparseChainComplexOfFilteredRegularCWComplex    Examples:

SparseChainComplexOfRegularCWComplexWithVectorField    Examples:

SparseChainComplexOfSimplicialComplex    Examples:

SparseChainComplexToChainComplex    Examples:

SparseChainMapOfCubicalPairs    Examples:

SparseFilteredChainComplexOfFilteredCubicalComplex    Examples:

SparseFilteredChainComplexOfFilteredSimplicialComplex    Examples: 1 , 2 

SparseMattoMat    Examples: 1 

SparseRowReduce    Examples:

SphericalKnotComplement    Examples: 1 

Spin    Examples:

SpunAboutHyperplane    Examples:

SpunKnotComplement    Examples: 1 

SpunLinkComplement    Examples:

StrongGeneratorsOfDerivedSubgroup    Examples:

StrongGeneratorsOfDerivedSubgroup_alt    Examples:

StructuralCopyOfFilteredRegularCWComplex    Examples:

SubQuasiIsomorph    Examples:

SubdivideCell    Examples:

Suspension_alt    Examples:

SylowSubgroupOfCatOneGroup    Examples:

SymmetricCentre    Examples:

SymmetricCommutativityGroup    Examples:

TensorNonFreeResolutionWithRationals    Examples:

TensorWithBurnsideRing    Examples: 1 , 2 

TensorWithComplexRepresentationRing    Examples: 1 , 2 

TensorWithComplexRepresentationRingOnRight    Examples:

TensorWithIntegersModPSparse    Examples:

TensorWithIntegersOverSubgroup    Examples: 1 , 2 , 3 , 4 

TensorWithIntegersSparse    Examples:

TensorWithModPModule    Examples: 1 

TestHapBook    Examples:

TestHapQuick    Examples:

ThickenedHEPureCubicalComplex    Examples:

ThickenedPureComplex    Examples: 1 

ThickenedPureCubicalComplex_dim2    Examples:

ThirdHomotopyGroupOfSuspensionB_alt    Examples: 1 

ThreeManifoldViaDehnSurgery    Examples: 1 

ThreeManifoldWithBoundary    Examples: 1 

TransferChainMap    Examples: 1 

TransferCochainMap    Examples: 1 

TranslationSubGroup    Examples:

TreeOfResolutionsToSL2Zcomplex    Examples:

TruncatedRegularCWComplex    Examples:

Tube    Examples:

TupleOrbitReps    Examples:

TupleOrbitReps_perm    Examples:

TwistedResolution    Examples:

UnboundedArrayAssign    Examples:

UnitBall    Examples:

UnitCubicalBall    Examples:

UnitPermutahedralBall    Examples:

UniversalBarCodeEval    Examples:

UniversalCover    Examples: 1 , 2 , 3 , 4 

VectorToCrystMatrix    Examples:

VectorsToOneSkeleton    Examples: 1 

VerticesOfRegularCWCell    Examples:

View3dPureComplex    Examples:

ViewArc2Presentation    Examples:

ViewPureComplex    Examples:

VirtuallySimplicialSubdivision    Examples:

WeakCommutativityGroup    Examples:

WirtingerGroup    Examples: 1 

WirtingerGroup_gc    Examples:

WordModP    Examples:

ZigZagContractedFilteredPureCubicalComplex    Examples:

ZigZagContractedPureComplex    Examples: 1 

Sq    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 

Category_Of_Groups    Examples: 1 

PreImagesElmNC    Examples:

PreImagesNC    Examples:

PreImagesSetNC    Examples:

AsFpGroup    Examples:

BarycentricSubdivision    Examples: 1 , 2 

Bockstein    Examples: 1 , 2 , 3 

CategoryArrow    Examples:

CategoryObject    Examples:

ClosedSurface    Examples: 1 

CoboundaryMatrix    Examples:

CoefficientsOfPoincareSeries    Examples:

CohomologyClass    Examples: 1 , 2 

CohomologyRing    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 

ComplexProjectiveSpace    Examples: 1 

CompositionRingHomomorphism    Examples:

ConnectedComponentsQuandle    Examples:

ConnectedSum    Examples: 1 , 2 

DegreeOfRepresentative    Examples:

Dimensions    Examples:

ExcisedPair    Examples:

ExpandedComplex    Examples: 1 

FilteredRegularCWComplex    Examples: 1 

FundamentalGroupWithPathReps    Examples: 1 , 2 

GDerivedSubgroup    Examples:

GModuleAsGOuterGroup    Examples: 1 , 2 , 3 , 4 

GOuterGroupHomomorphism    Examples: 1 , 2 

GOuterGroupHomomorphism    Examples: 1 , 2 

GradedAlgebraPresentation    Examples:

GradedAlgebraPresentationNC    Examples:

HAPDerivationNC    Examples:

HAPRingHomomorphismByIndeterminateMap    Examples:

HAPRingReductionHomomorphism    Examples:

HAPRingReductionHomomorphism    Examples:

HAPRingToSubringHomomorphism    Examples:

HAPSubringToRingHomomorphism    Examples:

HAPSubringToRingHomomorphism    Examples:

HAPZeroRingHomomorphism    Examples:

HAP_EquivalenceClasses    Examples:

HomomorphismsImages    Examples:

ImageOfDerivation    Examples:

ImageOfRingHomomorphism    Examples:

IsAssociatedGradedRing    Examples:

KernelOfDerivation    Examples:

LowerGCentralSeries    Examples:

PathComponents    Examples: 1 

PersistentBettiNumbersAlt    Examples: 1 

PersistentHomology    Examples: 1 , 2 

PersistentHomology    Examples: 1 , 2 

PersistentHomology    Examples: 1 , 2 

PoincareSeriesAutoMem    Examples:

PoincareSeriesAutoMem    Examples:

PoincareSeriesAutoMemStop    Examples:

PolynomialToRModuleRep    Examples:

PreimageOfRingHomomorphism    Examples:

PureComplexMeet    Examples:

PureComplexRandomCell    Examples: 1 

PureComplexSubcomplex    Examples:

Pushout    Examples:

QuadraticIdeal    Examples: 1 

ReduceIdeal    Examples:

ReducedPolynomialRingPresentation    Examples:

ReducedPolynomialRingPresentationMap    Examples:

RefinedColouring    Examples:

Resolution    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 

RightTransversal_alt    Examples:

RingOfIntegers    Examples: 1 

SingularPolynomialNormalForm    Examples:

SingularSetNormalFormIdeal    Examples:

SingularSetNormalFormIdealNC    Examples:

SparseChainComplexOfPair    Examples:

Sphere    Examples: 1 

Sq    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 

Standard2Cocycle    Examples:

Standard2Cocycle    Examples:

StandardNCocycle    Examples:

StandardNCocycle    Examples:

StarGraph    Examples:

SubspaceBasisRepsByDegree    Examples:

SubspaceDimensionDegree    Examples:

Suspension    Examples: 1 , 2 , 3 , 4 , 5 

TrivialGModuleAsGOuterGroup    Examples: 1 , 2 , 3 , 4 

VertexLink    Examples:

VertexStar    Examples:

WedgeSum    Examples: 1 

TensorProductOp    Examples:

Arity    Examples:

AssociatedNumberField    Examples:

AssociatedRing    Examples:

Base    Examples: 1 

BaseElement    Examples:

BaseRing    Examples:

Cocycle    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 

CoefficientModule    Examples:

CohomologicalPeriod    Examples: 1 

CoxeterMatrix    Examples: 1 

DerivationImages    Examples:

DerivationRelations    Examples:

DerivationRing    Examples:

Fibre    Examples:

FibreElement    Examples:

Generators    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 

GeneratorsOfPresentationIdeal    Examples:

GradedAlgebraPresentationFamily    Examples:

HAPDerivationFamily    Examples:

HAPPRIME_HilbertSeries    Examples:

HAPRingHomomorphismFamily    Examples:

HAP_MultiplicativeGenerators    Examples:

IdentityMap    Examples:

ImageGenerators    Examples:

ImagePolynomialRing    Examples:

ImageRelations    Examples:

InCcGroup    Examples:

IndeterminateAndExponentOfUnivariateMonomial    Examples:

IndeterminateDegrees    Examples:

IndeterminatesOfGradedAlgebraPresentation    Examples:

IndeterminatesOfPolynomial    Examples:

IndexInSL2O    Examples: 1 

InnerAutomorphismGroupQuandle    Examples:

InnerAutomorphismGroupQuandleAsPerm    Examples:

InverseRingHomomorphism    Examples:

IsConnected    Examples: 1 , 2 , 3 

IsHomogeneousQuandle    Examples:

IsLatinQuandle    Examples: 1 

MaximumDegreeForPresentation    Examples:

ModPRingBasisAsPolynomials    Examples:

ModPRingGeneratorDegrees    Examples:

ModPRingNiceBasis    Examples:

ModPRingNiceBasisAsPolynomials    Examples:

Module    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 

NormOfIdeal    Examples:

OuterAction    Examples:

OuterGroup    Examples: 1 , 2 , 3 , 4 

PresentationIdeal    Examples:

PresentationOfGradedStructureConstantAlgebra    Examples: 1 

Pullbacks    Examples:

Pushouts    Examples:

RightMultiplicationGroupOfQuandle    Examples: 1 , 2 , 3 

RightMultiplicationGroupOfQuandleAsPerm    Examples: 1 

SingularGroebnerBasis    Examples:

SingularReducedGroebnerBasis    Examples:

SourceGenerators    Examples:

SourcePolynomialRing    Examples:

SourceRelations    Examples:

StarGraphAttr    Examples:

TermsOfPolynomial    Examples:

UnivariateMonomialsOfMonomial    Examples:

CoefficientsRing    Examples:

ElementsFamily    Examples:

IndexInSL2Z    Examples:

Name    Examples: 1 , 2 , 3 , 4 , 5 , 6 

Order    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 

IsAbelianCategory    Examples:

IsAdditiveCategory    Examples:

IsCategoryName    Examples:

IsCcGroup    Examples:

IsCrystTranslationSubGroup    Examples:

IsGOuterGroup    Examples:

IsGOuterGroupHomomorphism    Examples:

IsGammaSubgroupInSL3Z    Examples:

IsHAPRationalMatrixGroup    Examples:

IsHAPRationalSpecialLinearGroup    Examples:

IsIdealOfQuadraticIntegers    Examples:

IsPeriodic    Examples: 1 , 2 

IsPseudoListWithFunction    Examples:

IsQuadraticNumberField    Examples:

IsRingOfQuadraticIntegers    Examples:

IsStandard2Cocycle    Examples:

IsStandardNCocycle    Examples:

IsCcElement    Examples:

IsGradedAlgebraPresentation    Examples:

IsHAPDerivation    Examples:

IsHAPRingHomomorphism    Examples:

IsHAPRingModIdealObj    Examples:

IsHapCatOneGroup    Examples:

IsHapCatOneGroupMorphism    Examples:

IsHapChainComplex    Examples:

IsHapChainMap    Examples:

IsHapCochainComplex    Examples:

IsHapCochainMap    Examples:

IsHapCommutativeDiagram    Examples:

IsHapConjQuandElt    Examples:

IsHapCrossedModule    Examples:

IsHapCrossedModuleMorphism    Examples:

IsHapCubicalComplex    Examples:

IsHapEquivariantCWComplex    Examples:

IsHapEquivariantChainComplex    Examples:

IsHapEquivariantChainMap    Examples:

IsHapEquivariantNonFreeChainComplex    Examples:

IsHapEquivariantSpectralSequencePage    Examples:

IsHapFilteredChainComplex    Examples:

IsHapFilteredCubicalComplex    Examples:

IsHapFilteredGraph    Examples:

IsHapFilteredPureCubicalComplex    Examples:

IsHapFilteredRegularCWComplex    Examples:

IsHapFilteredSimplicialComplex    Examples:

IsHapFilteredSparseChainComplex    Examples:

IsHapGCocomplex    Examples:

IsHapGComplex    Examples:

IsHapGComplexMap    Examples:

IsHapGraph    Examples:

IsHapOppositeElement    Examples:

IsHapPureCubicalComplex    Examples:

IsHapPureCubicalLink    Examples:

IsHapPurePermutahedralComplex    Examples:

IsHapQuandlePresentation    Examples:

IsHapQuotientElement    Examples:

IsHapRegularCWComplex    Examples:

IsHapRegularCWMap    Examples:

IsHapResolution    Examples:

IsHapSimplicialComplex    Examples:

IsHapSimplicialFreeAbelianGroup    Examples:

IsHapSimplicialGroup    Examples:

IsHapSimplicialGroupMorphism    Examples:

IsHapSimplicialMap    Examples:

IsHapSparseChainComplex    Examples:

IsHapSparseChainMap    Examples:

IsHapSparseMat    Examples:

IsHapTorsionSubcomplex    Examples:

IsPseudoList    Examples:

IsCcElementRep    Examples:

IsGradedAlgebraPresentationRep    Examples:

IsHAPDerivationRep    Examples:

IsHAPIdealRep    Examples:

IsHAPRingHomomorphismIndeterminateMapRep    Examples:

IsHAPRingReductionHomomorphismRep    Examples:

IsHAPRingToSubringHomomorphismRep    Examples:

IsHAPSubringToRingHomomorphismRep    Examples:

IsHAPZeroRingHomomorphismRep    Examples:

IsHapCatOneGroupMorphismRep    Examples:

IsHapCatOneGroupRep    Examples:

IsHapChainComplexRep    Examples:

IsHapChainMapRep    Examples:

IsHapCochainComplexRep    Examples:

IsHapCochainMapRep    Examples:

IsHapCommutativeDiagramRep    Examples:

IsHapConjQuandEltRep    Examples:

IsHapCrossedModuleMorphismRep    Examples:

IsHapCrossedModuleRep    Examples:

IsHapCubicalComplexRep    Examples:

IsHapEquivariantCWComplexRep    Examples:

IsHapEquivariantChainComplexRep    Examples:

IsHapEquivariantChainMapRep    Examples:

IsHapEquivariantNonFreeChainComplexRep    Examples:

IsHapEquivariantSpectralSequencePageRep    Examples:

IsHapFilteredChainComplexRep    Examples:

IsHapFilteredCubicalComplexRep    Examples:

IsHapFilteredGraphRep    Examples:

IsHapFilteredPureCubicalComplexRep    Examples:

IsHapFilteredRegularCWComplexRep    Examples:

IsHapFilteredSimplicialComplexRep    Examples:

IsHapFilteredSparseChainComplexRep    Examples:

IsHapGCocomplexRep    Examples:

IsHapGComplexMapRep    Examples:

IsHapGComplexRep    Examples:

IsHapGraphRep    Examples:

IsHapOppositeElementRep    Examples:

IsHapPureCubicalComplexRep    Examples:

IsHapPureCubicalLinkRep    Examples:

IsHapPurePermutahedralComplexRep    Examples:

IsHapQuandlePresentationRep    Examples:

IsHapQuotientElementRep    Examples:

IsHapRegularCWComplexRep    Examples:

IsHapRegularCWMapRep    Examples:

IsHapResolutionRep    Examples:

IsHapSimplicialComplexRep    Examples:

IsHapSimplicialFreeAbelianGroupRep    Examples:

IsHapSimplicialGroupMorphismRep    Examples:

IsHapSimplicialGroupRep    Examples:

IsHapSimplicialMapRep    Examples:

IsHapSparseChainComplexRep    Examples:

IsHapSparseChainMapRep    Examples:

IsHapSparseMatRep    Examples:

IsHapTorsionSubcomplexRep    Examples:

IsPseudoListRep    Examples:

IdealOfQuadraticIntegers    Examples:

QuadraticNF    Examples:

RingOfQuadraticIntegers    Examples:

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

+    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 

+    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 

+    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 

    Examples:

    Examples:

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

AdditiveInverseMutable    Examples:

AsFpGroup    Examples:

AsList    Examples:

AsSSortedList    Examples:

BarycentricSubdivision    Examples: 1 , 2 

BaseRing    Examples:

Bockstein    Examples: 1 , 2 , 3 

CategoryArrow    Examples:

CategoryObject    Examples:

CoboundaryMatrix    Examples:

CoefficientsOfPoincareSeries    Examples:

CoefficientsRing    Examples:

CohomologicalPeriod    Examples: 1 

CohomologyClass    Examples: 1 , 2 

CompositionRingHomomorphism    Examples:

CompositionRingHomomorphism    Examples:

CompositionRingHomomorphism    Examples:

CompositionRingHomomorphism    Examples:

CompositionRingHomomorphism    Examples:

ConnectedComponentsQuandle    Examples:

ConnectedSum    Examples: 1 , 2 

CoxeterMatrix    Examples: 1 

DefaultFieldOfMatrixGroup    Examples:

DegreeOfRepresentative    Examples:

DerivationImages    Examples:

DerivationRelations    Examples:

DerivationRing    Examples:

Dimensions    Examples:

Enumerator    Examples:

ExcisedPair    Examples:

FilteredRegularCWComplex    Examples: 1 

FundamentalGroupWithPathReps    Examples: 1 , 2 

GModuleAsGOuterGroup    Examples: 1 , 2 , 3 , 4 

GOuterGroupHomomorphism    Examples: 1 , 2 

GOuterGroupHomomorphism    Examples: 1 , 2 

Generators    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 

GeneratorsOfMagmaWithInverses    Examples:

GeneratorsOfMagmaWithInverses    Examples:

GeneratorsOfPresentationIdeal    Examples:

GradedAlgebraPresentation    Examples:

GradedAlgebraPresentationNC    Examples:

HAPDerivationNC    Examples:

HAPPRIME_HilbertSeries    Examples:

HAPRingHomomorphismByIndeterminateMap    Examples:

HAPRingReductionHomomorphism    Examples:

HAPRingReductionHomomorphism    Examples:

HAPRingToSubringHomomorphism    Examples:

HAPSubringToRingHomomorphism    Examples:

HAPSubringToRingHomomorphism    Examples:

HAPZeroRingHomomorphism    Examples:

HAP_MultiplicativeGenerators    Examples:

HomomorphismsImages    Examples:

IdGroup    Examples: 1 , 2 , 3 , 4 , 5 , 6 

IdentityMap    Examples:

ImageOfDerivation    Examples:

ImageOfRingHomomorphism    Examples:

ImageOfRingHomomorphism    Examples:

ImageOfRingHomomorphism    Examples:

ImageOfRingHomomorphism    Examples:

ImageOfRingHomomorphism    Examples:

IndeterminateAndExponentOfUnivariateMonomial    Examples:

IndeterminateDegrees    Examples:

IndeterminatesOfGradedAlgebraPresentation    Examples:

IndeterminatesOfPolynomial    Examples:

IndexInSL2O    Examples: 1 

IndexInSL2Z    Examples:

InnerAutomorphismGroupQuandle    Examples:

InnerAutomorphismGroupQuandleAsPerm    Examples:

Int    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

InverseMutable    Examples:

InverseMutable    Examples:

InverseMutable    Examples:

InverseRingHomomorphism    Examples:

InverseRingHomomorphism    Examples:

InverseRingHomomorphism    Examples:

InverseRingHomomorphism    Examples:

InverseRingHomomorphism    Examples:

InverseSameMutability    Examples:

IsAssociatedGradedRing    Examples:

IsConnected    Examples: 1 , 2 , 3 

IsHomogeneousQuandle    Examples:

IsLatinQuandle    Examples: 1 

IsMonomial    Examples:

IsOne    Examples:

IsPeriodic    Examples: 1 , 2 

Kernel    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 

KernelOfDerivation    Examples:

MaximumDegreeForPresentation    Examples:

ModPRingBasisAsPolynomials    Examples:

ModPRingGeneratorDegrees    Examples:

ModPRingNiceBasis    Examples:

ModPRingNiceBasisAsPolynomials    Examples:

OneImmutable    Examples:

OneMutable    Examples:

PathComponents    Examples: 1 

PersistentBettiNumbersAlt    Examples: 1 

PreimageOfRingHomomorphism    Examples:

PresentationIdeal    Examples:

PresentationOfGradedStructureConstantAlgebra    Examples: 1 

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

Projection    Examples:

PureComplexMeet    Examples:

PureComplexRandomCell    Examples: 1 

PureComplexSubcomplex    Examples:

Pushout    Examples:

QuadraticIdeal    Examples: 1 

Random    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 

ReduceIdeal    Examples:

ReducedPolynomialRingPresentation    Examples:

ReducedPolynomialRingPresentationMap    Examples:

RefinedColouring    Examples:

Resolution    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 

RightMultiplicationGroupOfQuandle    Examples: 1 , 2 , 3 

RightMultiplicationGroupOfQuandleAsPerm    Examples: 1 

RightTransversal    Examples:

RingOfIntegers    Examples: 1 

SingularGroebnerBasis    Examples:

SingularPolynomialNormalForm    Examples:

SingularReducedGroebnerBasis    Examples:

SingularSetNormalFormIdeal    Examples:

SingularSetNormalFormIdealNC    Examples:

SparseChainComplexOfPair    Examples:

Sq    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 

Standard2Cocycle    Examples:

Standard2Cocycle    Examples:

StandardNCocycle    Examples:

StandardNCocycle    Examples:

StarGraph    Examples:

StarGraphAttr    Examples:

SubspaceBasisRepsByDegree    Examples:

SubspaceDimensionDegree    Examples:

Suspension    Examples: 1 , 2 , 3 , 4 , 5 

TensorProductOp    Examples:

TermsOfPolynomial    Examples:

TrivialGModuleAsGOuterGroup    Examples: 1 , 2 , 3 , 4 

Units    Examples:

Units    Examples:

UnivariateMonomialsOfMonomial    Examples:

VertexLink    Examples:

VertexStar    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

WedgeSum    Examples: 1 

ZeroMutable    Examples:

^    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 

^    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 

^    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

mod    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

+    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 

+    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 

+    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 

-    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

-    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

/    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

    Examples:

    Examples:

    Examples:

    Examples:

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

AbelianInvariants    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 

AbelianInvariants    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 

BarycentricSubdivision    Examples: 1 , 2 

Bockstein    Examples: 1 , 2 , 3 

CanonicalRightCosetElement    Examples:

ClosedSurface    Examples: 1 

CohomologyRing    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 

CohomologyRing    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 

ComplexProjectiveSpace    Examples: 1 

ConnectedSum    Examples: 1 , 2 

ConnectedSum    Examples: 1 , 2 

ConnectedSum    Examples: 1 , 2 

Dimensions    Examples:

DirectProductOp    Examples:

DirectProductOp    Examples:

DirectProductOp    Examples:

DirectProductOp    Examples:

Discriminant    Examples:

Discriminant    Examples:

Embedding    Examples:

ExpandedComplex    Examples: 1 

ExpandedComplex    Examples: 1 

ExpandedComplex    Examples: 1 

ExpandedComplex    Examples: 1 

GDerivedSubgroup    Examples:

Generators    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 

HAPRingReductionHomomorphism    Examples:

HAPRingReductionHomomorphism    Examples:

HAP_EquivalenceClasses    Examples:

ImageOfRingHomomorphism    Examples:

ImageOfRingHomomorphism    Examples:

IndexNC    Examples:

IndexNC    Examples:

InverseMutable    Examples:

InverseMutable    Examples:

InverseMutable    Examples:

InverseSameMutability    Examples:

IsEmpty    Examples:

IsEmpty    Examples:

IsPrime    Examples: 1 , 2 

IsomorphismFpGroup    Examples: 1 , 2 

Iterator    Examples:

KernelOfDerivation    Examples:

ListOp    Examples:

ListOp    Examples:

LowerGCentralSeries    Examples:

NaturalHomomorphism    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 

Norm    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 

Norm    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 

OneImmutable    Examples:

OneImmutable    Examples:

Order    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 

Order    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 

PersistentBettiNumbersAlt    Examples: 1 

PersistentBettiNumbersAlt    Examples: 1 

PersistentBettiNumbersAlt    Examples: 1 

PersistentBettiNumbersAlt    Examples: 1 

PersistentBettiNumbersAlt    Examples: 1 

PersistentHomology    Examples: 1 , 2 

PersistentHomology    Examples: 1 , 2 

Position    Examples: 1 , 2 

Position    Examples: 1 , 2 

Position    Examples: 1 , 2 

PositionCanonical    Examples:

PreimageOfRingHomomorphism    Examples:

Projection    Examples:

PureComplexMeet    Examples:

PureComplexRandomCell    Examples: 1 

PureComplexSubcomplex    Examples:

QuadraticIdeal    Examples: 1 

Range    Examples: 1 , 2 

RankMatrixDestructive    Examples:

ReduceIdeal    Examples:

ReduceIdeal    Examples:

ReducedPolynomialRingPresentation    Examples:

ReducedPolynomialRingPresentation    Examples:

ReducedPolynomialRingPresentationMap    Examples:

ReducedPolynomialRingPresentationMap    Examples:

ReducedPolynomialRingPresentationMap    Examples:

RefinedColouring    Examples:

RightTransversal    Examples:

RightTransversal    Examples:

RightTransversal    Examples:

RightTransversal    Examples:

RightTransversal    Examples:

RightTransversal_alt    Examples:

SingularPolynomialNormalForm    Examples:

SparseChainComplexOfPair    Examples:

Sphere    Examples: 1 

SubspaceBasisRepsByDegree    Examples:

SubspaceDimensionDegree    Examples:

Suspension    Examples: 1 , 2 , 3 , 4 , 5 

TensorProductOp    Examples:

TensorProductOp    Examples:

TensorProductOp    Examples:

Trace    Examples:

Units    Examples:

WedgeSum    Examples: 1 

WedgeSum    Examples: 1 

WedgeSum    Examples: 1 

[]    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 

[]    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 

[]    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 

^    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 

^    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 

^    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 

^    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

mod    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 

PathComponentOfSimplicialComplex    Examples:

ResolutionSL2ZInvertedInteger    Examples:

ViewGraph    Examples:

InfoHAPprime    Examples:

ASY_PATH    Examples:

AutomorphismGroupAsCrossedModule    Examples:

BROWSER_PATH    Examples:

CATONEGROUP_DATA_PERM    Examples:

CATONEGROUP_DATA_SIZE    Examples:

Cedric_PlanarDiagram    Examples:

ChildKill    Examples:

DISPLAY_PATH    Examples:

DOT_PATH    Examples:

FilteredSimplicialComplexToFilteredCWComplex    Examples:

GradedAlgebraPresentationType    Examples:

HAPTEMPORARYFUNCTION    Examples:

HAP_Knots    Examples:

HAP_ROOT    Examples:

HapCatOneGroup    Examples:

HapCatOneGroupFamily    Examples:

HapCatOneGroupMorphism    Examples:

HapCatOneGroupMorphismFamily    Examples:

HapChainComplex    Examples:

HapChainComplexFamily    Examples:

HapChainMap    Examples:

HapChainMapFamily    Examples:

HapCochainComplex    Examples:

HapCochainComplexFamily    Examples:

HapCochainMap    Examples:

HapCochainMapFamily    Examples:

HapCommutativeDiagram    Examples:

HapCommutativeDiagramFamily    Examples:

HapCrossedModule    Examples:

HapCrossedModuleFamily    Examples:

HapCrossedModuleMorphism    Examples:

HapCrossedModuleMorphismFamily    Examples:

HapCubicalComplex    Examples:

HapCubicalComplexFamily    Examples:

HapEquivariantCWComplex    Examples:

HapEquivariantCWComplexFamily    Examples:

HapEquivariantChainMap    Examples:

HapEquivariantChainMapFamily    Examples:

HapFPGModule    Examples:

HapFPGModuleHomomorphism    Examples:

HapFilteredChainComplex    Examples:

HapFilteredChainComplexFamily    Examples:

HapFilteredCubicalComplex    Examples:

HapFilteredCubicalComplexFamily    Examples:

HapFilteredGraph    Examples:

HapFilteredGraphFamily    Examples:

HapFilteredPureCubicalComplex    Examples:

HapFilteredPureCubicalComplexFamily    Examples:

HapFilteredRegularCWComplex    Examples:

HapFilteredRegularCWComplexFamily    Examples:

HapFilteredSimplicialComplex    Examples:

HapFilteredSimplicialComplexFamily    Examples:

HapFilteredSparseChainComplex    Examples:

HapFilteredSparseChainComplexFamily    Examples:

HapGChainComplex    Examples:

HapGCocomplex    Examples:

HapGCocomplexFamily    Examples:

HapGComplex    Examples:

HapGComplexFamily    Examples:

HapGlobalDeclarationsAreAlreadyLoaded    Examples:

HapGraph    Examples:

HapGraphFamily    Examples:

HapNonFreeResolution    Examples:

HapOppositeElement    Examples:

HapOppositeElementFamily    Examples:

HapPureCubicalComplex    Examples:

HapPureCubicalComplexFamily    Examples:

HapPureCubicalLink    Examples:

HapPureCubicalLinkFamily    Examples:

HapPurePermutahedralComplex    Examples:

HapPurePermutahedralComplexFamily    Examples:

HapQuotientElement    Examples:

HapQuotientElementFamily    Examples:

HapRegularCWComplex    Examples:

HapRegularCWComplexFamily    Examples:

HapRegularCWMap    Examples:

HapRegularCWMapFamily    Examples:

HapResolution    Examples:

HapResolutionFamily    Examples:

HapSimplicialComplex    Examples:

HapSimplicialComplexFamily    Examples:

HapSimplicialGroup    Examples:

HapSimplicialGroupFamily    Examples:

HapSimplicialGroupMorphism    Examples:

HapSimplicialGroupMorphismFamily    Examples:

HapSimplicialMap    Examples:

HapSimplicialMapFamily    Examples:

HapSparseChainComplex    Examples:

HapSparseChainComplexFamily    Examples:

HapSparseChainMap    Examples:

HapSparseChainMapFamily    Examples:

HapSparseMat    Examples:

HapSparseMatFamily    Examples:

HomomorphismOfDirectProduct    Examples:

IDQUASICATONEGROUP_DATA    Examples:

IsHapChain    Examples:

IsHapCochain    Examples:

IsHapComplex    Examples:

IsHapFPGModule    Examples:

IsHapFPGModuleHomomorphism    Examples:

IsHapGChainComplex    Examples:

IsHapMap    Examples:

IsHapNonFreeResolution    Examples:

NEATO_PATH    Examples:

NerveOfCover    Examples:

POLYMAKE_PATH    Examples:

PseudoList    Examples:

PseudoListFamily    Examples:

QUASICATONEGROUP_DATA_NOT    Examples:

QUASICATONEGROUP_DATA_SIZE    Examples:

ReadBioData    Examples:

SMALLQUASICATONEGROUP_DATA    Examples:

CATONEGROUP_DATA    Examples:

COMPILED    Examples:

Cedric_XYXYConnQuan    Examples:

Cedric_XYXYQuandles    Examples:

CommutingProbability    Examples:

GroupIsomorphismRepresentatives    Examples:

HAPAAA    Examples:

HAPBARCODE    Examples:

HAPDerivationType    Examples:

HAPPRIME_LastLHSBicomplexSize    Examples:

HAPPRIME_ShuffleRandomSource    Examples:

HAPRIGXXX    Examples:

HAP_GCOMPLEX_LIST    Examples:

HAP_GCOMPLEX_SETUP    Examples:

HAP_MOVES_DIM_2    Examples:

HAP_MOVES_DIM_3    Examples:

HAP_PERMMOVES_DIM_2    Examples:

HAP_PERMMOVES_DIM_3    Examples:

HAP_PoincareCubeManifoldEdgeDegrees    Examples:

HAP_Test    Examples:

HAP_XYXYXYXY    Examples:

HAPchildFunctionToggle    Examples:

HAPchildToggle    Examples:

HAPchildren    Examples:

HapConjQuandElt    Examples:

HapConjQuandEltFamily    Examples:

HapConstantPolRing    Examples:

HapEquivariantChainComplex    Examples:

HapEquivariantChainComplexFamily    Examples:

HapEquivariantNonFreeChainComplex    Examples:

HapEquivariantNonFreeChainComplexFamily    Examples:

HapEquivariantSpectralSequencePage    Examples:

HapEquivariantSpectralSequencePageFamily    Examples:

HapGComplexMap    Examples:

HapGComplexMapFamily    Examples:

HapQuandlePresentation    Examples:

HapQuandlePresentationFamily    Examples:

HapRightTransversalSL2ZSubgroup    Examples:

HapSL2ZConjugatedSubgroup    Examples:

HapSL2ZSubgroup    Examples:

HapSimplicialFreeAbelianGroup    Examples:

HapSimplicialFreeAbelianGroupFamily    Examples:

HapTorsionSubcomplex    Examples:

HapTorsionSubcomplexFamily    Examples:

IntersectionForm    Examples: 1 , 2 

IsHapRightTransversalSL2ZSubgroup    Examples:

IsHapSL2ConjugatedSubgroup    Examples:

IsHapSL2OSubgroup    Examples:

IsHapSL2Subgroup    Examples:

IsHapSL2ZConjugatedSubgroup    Examples:

IsHapSL2ZSubgroup    Examples:

RefinedColouring_gc    Examples:

RefinedColouring_group    Examples:

RegularCWAssociahedronWithDiscreteVectorField    Examples:

RegularCWClosedSurface    Examples:

RegularCWComplexWithAttachedRelatorCells    Examples: 1 

RegularCWComplex_DisjointUnion    Examples:

RegularCWComplex_WedgeSum    Examples:

RegularCWDiscreteSpace    Examples: 1 

RegularCWSphere    Examples: 1 

SimplicialComplexConnectedSum    Examples:

SphericalKnotComplementWithBoundary    Examples:

StemGroups    Examples:

cat    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 

cnt    Examples:

hap_cr    Examples:

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diff --git a/doc/chap40.txt b/doc/chap40.txt index 0c61860f..804abfb3 100644 --- a/doc/chap40.txt +++ b/doc/chap40.txt @@ -80,6 +80,7 @@ ChainComplexWithChainHomotopy    Examples: ChainMapOfCubicalPairs    Examples: ChainMapOfRegularCWMap    Examples: + ChevalleyEilenbergComplexOfModule    Examples: ChildRestart    Examples: ClosureCWCell    Examples: CoClass    Examples: @@ -198,6 +199,7 @@ FiltrationTermOfGraph    Examples: FiltrationTermOfPureCubicalComplex    Examples: FiltrationTermOfRegularCWComplex    Examples: + FiltrationTerms    Examples: 1 (../tutorial/chap5.html)  FirstHomologyCoveringCokernels    Examples: 1 (../tutorial/chap3.html) , 2 (../www/SideLinks/About/aboutCoverinSpaces.html)  FirstHomologySimplicialTwoComplex    Examples: @@ -377,8 +379,8 @@ IdQuasiCatOneGroup    Examples: IdQuasiCrossedModule    Examples: IdentifyKnot    Examples: 1 (../tutorial/chap6.html)  - IdentityAmongRelators    Examples: 1 - (../www/SideLinks/About/aboutPeriodic.html) , 2 + IdentityAmongRelators    Examples: 1 (../tutorial/chap6.html) , 2 + (../www/SideLinks/About/aboutPeriodic.html) , 3 (../www/SideLinks/About/aboutTopology.html)  ImageOfGOuterGroupHomomorphism    Examples: 1 (../tutorial/chap7.html) , 2 (../www/SideLinks/About/aboutCoefficientSequence.html)  @@ -506,7 +508,8 @@ PoincareBipyramidCWComplex    Examples: 1 (../tutorial/chap4.html)  PoincareCubeCWComplex    Examples: 1 (../tutorial/chap4.html)  PoincareCubeCWComplexNS    Examples: 1 (../tutorial/chap4.html)  - PoincareDodecahedronCWComplex    Examples: 1 (../tutorial/chap4.html)  + PoincareDodecahedronCWComplex    Examples: 1 (../tutorial/chap4.html) , 2 + (../tutorial/chap11.html)  PoincareOctahedronCWComplex    Examples: 1 (../tutorial/chap4.html)  PoincarePrismCWComplex    Examples: 1 (../tutorial/chap4.html)  PoincareSeriesApproximation    Examples: @@ -692,22 +695,21 @@ ZigZagContractedFilteredPureCubicalComplex    Examples: ZigZagContractedPureComplex    Examples: 1 (../www/SideLinks/About/aboutPeripheral.html)  - Sq    Examples: 1 (../tutorial/chap8.html) , 2 (../tutorial/chap11.html) , 3 - (../www/SideLinks/About/aboutArtinGroups.html) , 4 - (../www/SideLinks/About/aboutModPRings.html) , 5 - (../www/SideLinks/About/aboutAspherical.html) , 6 - (../www/SideLinks/About/aboutNonabelian.html) , 7 - (../www/SideLinks/About/aboutQuandles2.html) , 8 - (../www/SideLinks/About/aboutKnots.html) , 9 - (../www/SideLinks/About/aboutTensorSquare.html) , 10 + Sq    Examples: 1 (../tutorial/chap5.html) , 2 (../tutorial/chap8.html) , 3 + (../tutorial/chap11.html) , 4 + (../www/SideLinks/About/aboutArtinGroups.html) , 5 + (../www/SideLinks/About/aboutModPRings.html) , 6 + (../www/SideLinks/About/aboutAspherical.html) , 7 + (../www/SideLinks/About/aboutNonabelian.html) , 8 + (../www/SideLinks/About/aboutQuandles2.html) , 9 + (../www/SideLinks/About/aboutKnots.html) , 10 + (../www/SideLinks/About/aboutTensorSquare.html) , 11 (../www/SideLinks/About/aboutKnotsQuandles.html)  Category_Of_Groups    Examples: 1 (../www/SideLinks/About/aboutAbelianCategories.html)  - ElementsSL2Z    Examples: - HAP_knot_census    Examples: - PathComponentOfSimplicialComplex    Examples: - ResolutionSL2ZInvertedInteger    Examples: - ViewGraph    Examples: + PreImagesElmNC    Examples: + PreImagesNC    Examples: + PreImagesSetNC    Examples: AsFpGroup    Examples: BarycentricSubdivision    Examples: 1 (../tutorial/chap1.html) , 2 (../tutorial/chap4.html)  @@ -735,6 +737,7 @@ DegreeOfRepresentative    Examples: Dimensions    Examples: ExcisedPair    Examples: + ExpandedComplex    Examples: 1 (../tutorial/chap5.html)  FilteredRegularCWComplex    Examples: 1 (../tutorial/chap5.html)  FundamentalGroupWithPathReps    Examples: 1 (../tutorial/chap1.html) , 2 (../tutorial/chap4.html)  @@ -778,7 +781,7 @@ PolynomialToRModuleRep    Examples: PreimageOfRingHomomorphism    Examples: PureComplexMeet    Examples: - PureComplexRandomCell    Examples: + PureComplexRandomCell    Examples: 1 (../tutorial/chap5.html)  PureComplexSubcomplex    Examples: Pushout    Examples: QuadraticIdeal    Examples: 1 (../tutorial/chap11.html)  @@ -824,14 +827,15 @@ SingularSetNormalFormIdealNC    Examples: SparseChainComplexOfPair    Examples: Sphere    Examples: 1 (../tutorial/chap1.html)  - Sq    Examples: 1 (../tutorial/chap8.html) , 2 (../tutorial/chap11.html) , 3 - (../www/SideLinks/About/aboutArtinGroups.html) , 4 - (../www/SideLinks/About/aboutModPRings.html) , 5 - (../www/SideLinks/About/aboutAspherical.html) , 6 - (../www/SideLinks/About/aboutNonabelian.html) , 7 - (../www/SideLinks/About/aboutQuandles2.html) , 8 - (../www/SideLinks/About/aboutKnots.html) , 9 - (../www/SideLinks/About/aboutTensorSquare.html) , 10 + Sq    Examples: 1 (../tutorial/chap5.html) , 2 (../tutorial/chap8.html) , 3 + (../tutorial/chap11.html) , 4 + (../www/SideLinks/About/aboutArtinGroups.html) , 5 + (../www/SideLinks/About/aboutModPRings.html) , 6 + (../www/SideLinks/About/aboutAspherical.html) , 7 + (../www/SideLinks/About/aboutNonabelian.html) , 8 + (../www/SideLinks/About/aboutQuandles2.html) , 9 + (../www/SideLinks/About/aboutKnots.html) , 10 + (../www/SideLinks/About/aboutTensorSquare.html) , 11 (../www/SideLinks/About/aboutKnotsQuandles.html)  Standard2Cocycle    Examples: Standard2Cocycle    Examples: @@ -1650,38 +1654,38 @@ InnerAutomorphismGroupQuandle    Examples: InnerAutomorphismGroupQuandleAsPerm    Examples: Int    Examples: 1 (../tutorial/chap1.html) , 2 (../tutorial/chap3.html) , 3 - (../tutorial/chap4.html) , 4 (../tutorial/chap7.html) , 5 - (../tutorial/chap8.html) , 6 (../tutorial/chap10.html) , 7 - (../tutorial/chap11.html) , 8 - (../www/SideLinks/About/aboutArithmetic.html) , 9 - (../www/SideLinks/About/aboutArtinGroups.html) , 10 - (../www/SideLinks/About/aboutAspherical.html) , 11 - (../www/SideLinks/About/aboutBogomolov.html) , 12 - (../www/SideLinks/About/aboutParallel.html) , 13 - (../www/SideLinks/About/aboutPerformance.html) , 14 - (../www/SideLinks/About/aboutCocycles.html) , 15 - (../www/SideLinks/About/aboutCohomologyRings.html) , 16 - (../www/SideLinks/About/aboutPersistent.html) , 17 - (../www/SideLinks/About/aboutPoincareSeries.html) , 18 - (../www/SideLinks/About/aboutCoveringSpaces.html) , 19 - (../www/SideLinks/About/aboutCoverinSpaces.html) , 20 - (../www/SideLinks/About/aboutPolytopes.html) , 21 - (../www/SideLinks/About/aboutCoxeter.html) , 22 - (../www/SideLinks/About/aboutCrossedMods.html) , 23 - (../www/SideLinks/About/aboutRosenbergerMonster.html) , 24 - (../www/SideLinks/About/aboutDavisComplex.html) , 25 - (../www/SideLinks/About/aboutSchurMultiplier.html) , 26 - (../www/SideLinks/About/aboutDefinitions.html) , 27 - (../www/SideLinks/About/aboutSimplicialGroups.html) , 28 - (../www/SideLinks/About/aboutExtensions.html) , 29 - (../www/SideLinks/About/aboutSpaceGroup.html) , 30 - (../www/SideLinks/About/aboutFunctorial.html) , 31 - (../www/SideLinks/About/aboutGraphsOfGroups.html) , 32 - (../www/SideLinks/About/aboutTDA.html) , 33 - (../www/SideLinks/About/aboutIntro.html) , 34 - (../www/SideLinks/About/aboutTensorSquare.html) , 35 - (../www/SideLinks/About/aboutTorAndExt.html) , 36 - (../www/SideLinks/About/aboutLie.html) , 37 + (../tutorial/chap4.html) , 4 (../tutorial/chap5.html) , 5 + (../tutorial/chap7.html) , 6 (../tutorial/chap8.html) , 7 + (../tutorial/chap10.html) , 8 (../tutorial/chap11.html) , 9 + (../www/SideLinks/About/aboutArithmetic.html) , 10 + (../www/SideLinks/About/aboutArtinGroups.html) , 11 + (../www/SideLinks/About/aboutAspherical.html) , 12 + (../www/SideLinks/About/aboutBogomolov.html) , 13 + (../www/SideLinks/About/aboutParallel.html) , 14 + (../www/SideLinks/About/aboutPerformance.html) , 15 + (../www/SideLinks/About/aboutCocycles.html) , 16 + (../www/SideLinks/About/aboutCohomologyRings.html) , 17 + (../www/SideLinks/About/aboutPersistent.html) , 18 + (../www/SideLinks/About/aboutPoincareSeries.html) , 19 + (../www/SideLinks/About/aboutCoveringSpaces.html) , 20 + (../www/SideLinks/About/aboutCoverinSpaces.html) , 21 + (../www/SideLinks/About/aboutPolytopes.html) , 22 + (../www/SideLinks/About/aboutCoxeter.html) , 23 + (../www/SideLinks/About/aboutCrossedMods.html) , 24 + (../www/SideLinks/About/aboutRosenbergerMonster.html) , 25 + (../www/SideLinks/About/aboutDavisComplex.html) , 26 + (../www/SideLinks/About/aboutSchurMultiplier.html) , 27 + (../www/SideLinks/About/aboutDefinitions.html) , 28 + (../www/SideLinks/About/aboutSimplicialGroups.html) , 29 + (../www/SideLinks/About/aboutExtensions.html) , 30 + (../www/SideLinks/About/aboutSpaceGroup.html) , 31 + (../www/SideLinks/About/aboutFunctorial.html) , 32 + (../www/SideLinks/About/aboutGraphsOfGroups.html) , 33 + (../www/SideLinks/About/aboutTDA.html) , 34 + (../www/SideLinks/About/aboutIntro.html) , 35 + (../www/SideLinks/About/aboutTensorSquare.html) , 36 + (../www/SideLinks/About/aboutTorAndExt.html) , 37 + (../www/SideLinks/About/aboutLie.html) , 38 (../www/SideLinks/About/aboutTwistedCoefficients.html)  InverseMutable    Examples: InverseMutable    Examples: @@ -1795,7 +1799,7 @@ PrintObj    Examples: Projection    Examples: PureComplexMeet    Examples: - PureComplexRandomCell    Examples: + PureComplexRandomCell    Examples: 1 (../tutorial/chap5.html)  PureComplexSubcomplex    Examples: Pushout    Examples: QuadraticIdeal    Examples: 1 (../tutorial/chap11.html)  @@ -1861,14 +1865,15 @@ SingularSetNormalFormIdeal    Examples: SingularSetNormalFormIdealNC    Examples: SparseChainComplexOfPair    Examples: - Sq    Examples: 1 (../tutorial/chap8.html) , 2 (../tutorial/chap11.html) , 3 - (../www/SideLinks/About/aboutArtinGroups.html) , 4 - (../www/SideLinks/About/aboutModPRings.html) , 5 - (../www/SideLinks/About/aboutAspherical.html) , 6 - (../www/SideLinks/About/aboutNonabelian.html) , 7 - (../www/SideLinks/About/aboutQuandles2.html) , 8 - (../www/SideLinks/About/aboutKnots.html) , 9 - (../www/SideLinks/About/aboutTensorSquare.html) , 10 + Sq    Examples: 1 (../tutorial/chap5.html) , 2 (../tutorial/chap8.html) , 3 + (../tutorial/chap11.html) , 4 + (../www/SideLinks/About/aboutArtinGroups.html) , 5 + (../www/SideLinks/About/aboutModPRings.html) , 6 + (../www/SideLinks/About/aboutAspherical.html) , 7 + (../www/SideLinks/About/aboutNonabelian.html) , 8 + (../www/SideLinks/About/aboutQuandles2.html) , 9 + (../www/SideLinks/About/aboutKnots.html) , 10 + (../www/SideLinks/About/aboutTensorSquare.html) , 11 (../www/SideLinks/About/aboutKnotsQuandles.html)  Standard2Cocycle    Examples: Standard2Cocycle    Examples: @@ -3325,6 +3330,10 @@ Discriminant    Examples: Discriminant    Examples: Embedding    Examples: + ExpandedComplex    Examples: 1 (../tutorial/chap5.html)  + ExpandedComplex    Examples: 1 (../tutorial/chap5.html)  + ExpandedComplex    Examples: 1 (../tutorial/chap5.html)  + ExpandedComplex    Examples: 1 (../tutorial/chap5.html)  GDerivedSubgroup    Examples: Generators    Examples: 1 (../tutorial/chap1.html) , 2 (../tutorial/chap3.html) , 3 (../tutorial/chap4.html) , 4 @@ -3466,7 +3475,7 @@ PreimageOfRingHomomorphism    Examples: Projection    Examples: PureComplexMeet    Examples: - PureComplexRandomCell    Examples: + PureComplexRandomCell    Examples: 1 (../tutorial/chap5.html)  PureComplexSubcomplex    Examples: QuadraticIdeal    Examples: 1 (../tutorial/chap11.html)  Range    Examples: 1 (../tutorial/chap7.html) , 2 @@ -3995,6 +4004,9 @@ (../www/SideLinks/About/aboutTopology.html) , 38 (../www/SideLinks/About/aboutTorAndExt.html) , 39 (../www/SideLinks/About/aboutTwistedCoefficients.html)  + PathComponentOfSimplicialComplex    Examples: + ResolutionSL2ZInvertedInteger    Examples: + ViewGraph    Examples: InfoHAPprime    Examples: ASY_PATH    Examples: AutomorphismGroupAsCrossedModule    Examples: diff --git a/doc/chap40_mj.html b/doc/chap40_mj.html index 9cfd3015..bc6c976e 100644 --- a/doc/chap40_mj.html +++ b/doc/chap40_mj.html @@ -36,7 +36,7 @@

40 HAP variables that are not yet documented40.1  

-

2CoreducedChainComplex    Examples:

AbelianGOuterGroupToCatOneGroup    Examples:

AbelianInvariantsToTorsionCoefficients    Examples:

AcyclicSubcomplexOfPureCubicalComplex    Examples: 1 

AddFirst    Examples:

AdjointGroupOfQuandle    Examples: 1 

AlgebraicReduction_alt    Examples:

AppendFreeWord    Examples:

ArcDiagramToTubularSurface    Examples:

ArcPresentation    Examples: 1 , 2 , 3 , 4 

ArcPresentationToKnottedOneComplex    Examples:

AreIsoclinic    Examples:

ArrayIterateBreak    Examples:

ArrayValueKD    Examples:

AsWordInSL2Z    Examples:

AutomorphismGroupQuandleAsPerm_nonconnected    Examples:

AverageInnerProduct    Examples:

BarCodeOfFilteredPureCubicalComplex    Examples:

BarCodeOfSymmetricMatrix    Examples:

BarComplexOfMonoid    Examples: 1 

BarycentricallySimplifiedComplex    Examples: 1 

BarycentricallySubdivideCell    Examples:

BettinumbersOfPureCubicalComplex_dim_2    Examples:

BocksteinHomology    Examples:

BogomolovMultiplier_viaTensorSquare    Examples:

BoundariesOfFilteredChainComplex    Examples:

BoundaryOfPureComplex    Examples: 1 

BoundaryOfPureRegularCWComplex    Examples: 1 

BoundaryOfRegularCWCell    Examples:

BoundaryPairOfPureRegularCWComplex    Examples:

BoundingPureComplex    Examples:

CR_ChainMapFromCocycle    Examples:

CR_CocyclesAndCoboundaries    Examples:

CR_IntegralClassToCocycle    Examples:

CR_IntegralCocycleToClass    Examples:

CR_IntegralCohomology    Examples:

CR_IntegralCycleToClass    Examples:

CWMap2ChainMap    Examples:

CWSubcomplexToRegularCWMap    Examples: 1 

CanonicalRightCountableCosetElement    Examples:

CatOneGroupByCrossedModule    Examples:

CatOneGroupsByGroup    Examples:

CcElement    Examples:

Cedric_CheckThirdAxiomRow    Examples:

Cedric_ConjugateQuandleElement    Examples:

Cedric_FromAutGeReToAutQe    Examples:

Cedric_IsHomomorphism    Examples:

Cedric_Permute    Examples:

Cedric_Quandle1    Examples:

Cedric_Quandle2    Examples:

Cedric_Quandle3    Examples:

Cedric_Quandle4    Examples:

Cedric_Quandle5    Examples:

Cedric_Quandle6    Examples:

CellComplexBoundaryCheck    Examples:

ChainComplexEquivalenceOfRegularCWComplex    Examples: 1 

ChainComplexHomeomorphismEquivalenceOfRegularCWComplex    Examples:

ChainComplexOfCubicalComplex    Examples:

ChainComplexOfCubicalPair    Examples:

ChainComplexOfRegularCWComplexWithVectorField    Examples:

ChainComplexOfSimplicialComplex    Examples:

ChainComplexOfSimplicialPair    Examples:

ChainComplexOfUniversalCover    Examples: 1 , 2 , 3 , 4 

ChainComplexToSparseChainComplex    Examples:

ChainComplexWithChainHomotopy    Examples:

ChainMapOfCubicalPairs    Examples:

ChainMapOfRegularCWMap    Examples:

ChildRestart    Examples:

ClosureCWCell    Examples:

CoClass    Examples:

CocriticalCellsOfRegularCWComplex    Examples:

CocyclicHadamardMatrices    Examples: 1 

CocyclicMatrices    Examples:

CohomologicalData    Examples: 1 

CohomologyHomomorphism    Examples: 1 , 2 

CohomologyHomomorphismOfRepresentation    Examples:

CohomologyModule_AsAutModule    Examples:

CohomologyModule_Gmap    Examples:

CohomologyRingOfSimplicialComplex    Examples:

CohomologySimplicialFreeAbelianGroup    Examples:

CombinationDisjointSets    Examples:

CommonEndomorphisms    Examples:

ComplementOfPureComplex    Examples: 1 

ComplementaryBasis    Examples:

ComposeCWMaps    Examples:

CompositionOfFpGModuleHomomorphisms    Examples:

CompositionSeriesOfFpGModule    Examples:

ConcentricallyFilteredPureCubicalComplex    Examples: 1 

CongruenceSubgroup    Examples: 1 , 2 

ConjugateSL2ZGroup    Examples:

ConnectingCohomologyHomomorphism    Examples: 1 , 2 

ContractArray    Examples:

ContractCubicalComplex_dim2    Examples:

ContractCubicalComplex_dim3    Examples:

ContractMatrix    Examples:

ContractPermArray    Examples:

ContractPermMatrix    Examples:

ContractPureComplex    Examples:

ContractSimplicialComplex    Examples:

ContractSimplicialComplex_alt    Examples:

ContractedFilteredPureCubicalComplex    Examples: 1 

ContractedFilteredRegularCWComplex    Examples:

ContractedRegularCWComplex    Examples:

ContractibleSL2ZComplex    Examples:

ContractibleSL2ZComplex_alt    Examples:

ContractibleSubArray    Examples:

ContractibleSubMatrix    Examples:

ContractibleSubcomplexOfPureCubicalComplex    Examples: 1 

ConvertTorsionComplexToGcomplex    Examples:

CosetsQuandle    Examples:

CountingCellsOfBaryCentricSubdivision    Examples:

CountingNumberOfCellsInBaryCentricSubdivision    Examples:

CoxeterComplex_alt    Examples: 1 

CoxeterDiagramMatCoxeterGroup    Examples:

CoxeterWythoffComplex    Examples:

CreateCoxeterMatrix    Examples: 1 

CriticalBoundaryCells    Examples: 1 

CropPureComplex    Examples:

CrossedInvariant    Examples:

CrossedModuleByAutomorphismGroup    Examples:

CrossedModuleByCatOneGroup    Examples:

CrossedModuleByNormalSubgroup    Examples: 1 

CrystCubicalTiling    Examples:

CrystFinitePartOfMatrix    Examples:

CrystGFullBasis    Examples: 1 , 2 

CrystGcomplex    Examples: 1 , 2 

CrystMatrix    Examples:

CrystTranslationMatrixToVector    Examples:

CrystallographicComplex    Examples:

CubicalToPermutahedralArray    Examples:

CupProductMatrix    Examples:

CupProductOfRegularCWComplex    Examples: 1 

CupProductOfRegularCWComplex_alt    Examples: 1 

CuspidalCohomologyHomomorphism    Examples:

CyclesOfFilteredChainComplex    Examples:

DavisComplex    Examples: 1 , 2 , 3 , 4 

DeformationRetract    Examples:

DensityMat    Examples:

DerivedGroupOfQuandle    Examples: 1 

DiagonalChainMap    Examples:

DijkgraafWittenInvariant    Examples: 1 

DirectProductOfGroupHomomorphisms    Examples:

DirectProductOfRegularCWComplexes    Examples:

DirectProductOfRegularCWComplexesLazy    Examples:

DirectProductOfSimplicialComplexes    Examples:

DisplayCSVknotFile    Examples:

DisplayVectorField    Examples:

E1CohomologyPage    Examples:

E1HomologyPage    Examples:

EilenbergMacLaneSimplicialFreeAbelianGroup    Examples:

ElementsLazy    Examples:

EquivariantCWComplexToRegularCWComplex    Examples: 1 , 2 , 3 , 4 

EquivariantCWComplexToRegularCWMap    Examples: 1 , 2 , 3 

EquivariantCWComplexToResolution    Examples:

ExcisedPureCubicalPair_dim_2    Examples:

ExtractTorsionSubcomplex    Examples:

FactorizationNParts    Examples:

FilteredChainComplexToFilteredSparseChainComplex    Examples:

FilteredCubicalComplexToFilteredRegularCWComplex    Examples: 1 

FilteredPureCubicalComplexToCubicalComplex    Examples: 1 

FiltrationTermOfGraph    Examples:

FiltrationTermOfPureCubicalComplex    Examples:

FiltrationTermOfRegularCWComplex    Examples:

FirstHomologyCoveringCokernels    Examples: 1 , 2 

FirstHomologySimplicialTwoComplex    Examples:

FourthHomotopyGroupOfDoubleSuspensionB    Examples:

Fp2PcpAbelianGroupHomomorphism    Examples:

FpGModuleSection    Examples:

FreeZGResolution    Examples:

FundamentalGroupOfRegularCWComplex    Examples: 1 

FundamentalGroupOfRegularCWMap    Examples:

FundamentalGroupSimplicialTwoComplex    Examples:

FundamentalMultiplesOfStiefelWhitneyClasses    Examples:

GChainComplex    Examples: 1 

GModuleAsCatOneGroup    Examples:

GammaSubgroupInSL3Z    Examples:

GaussCodeOfPureCubicalKnot    Examples: 1 , 2 , 3 , 4 

GetTorsionPowerSubcomplex    Examples:

GetTorsionSubcomplex    Examples:

GraphOfRegularCWComplex    Examples:

GraphOfResolutionsTest    Examples:

GraphOfResolutionsToGroups    Examples:

GroupHomomorphismToMatrix    Examples:

HAPCocontractRegularCWComplex    Examples:

HAPContractFilteredRegularCWComplex    Examples:

HAPContractRegularCWComplex    Examples:

HAPContractRegularCWComplex_Alt    Examples:

HAPPRIME_Algebra2Polynomial    Examples:

HAPPRIME_CohomologyRingWithoutResolution    Examples:

HAPPRIME_CombineIndeterminateMaps    Examples:

HAPPRIME_GradedAlgebraPresentationAvoidingIndeterminates    Examples:

HAPPRIME_LHSSpectralSequence    Examples:

HAPPRIME_MakeEliminationOrdering    Examples:

HAPPRIME_MapPolynomialIndeterminates    Examples:

HAPPRIME_Polynomial2Algebra    Examples:

HAPPRIME_RingHomomorphismsAreComposable    Examples:

HAPPRIME_SModule    Examples:

HAPPRIME_SingularGroebnerBasis    Examples:

HAPPRIME_SingularReducedGroebnerBasis    Examples:

HAPPRIME_SwitchGradedAlgebraRing    Examples:

HAPPRIME_SwitchPolynomialIndeterminates    Examples:

HAPPRIME_VersionWithSVN    Examples:

HAPRegularCWComplex    Examples:

HAPRegularCWPolytope    Examples:

HAPRemoveCellFromRegularCWComplex    Examples:

HAPRemoveVectorField    Examples:

HAPRingModIdeal    Examples:

HAPRingModIdealObj    Examples:

HAPTietzeReduction_Inf    Examples:

HAPTietzeReduction_OneLevel    Examples:

HAPTietzeReduction_OneStep    Examples:

HAP_4x4MatTo2x2Mat    Examples:

HAP_AddGenerator    Examples:

HAP_AllHomomorphisms    Examples:

HAP_AppendTo    Examples:

HAP_AssociahedronBoundaries    Examples:

HAP_AssociahedronCells    Examples:

HAP_BaryCentricSubdivisionGComplex    Examples:

HAP_BaryCentricSubdivisionRegularCWComplex    Examples:

HAP_Binlisttoint    Examples:

HAP_ChainComplexToEquivariantChainComplex    Examples:

HAP_CocyclesAndCoboundaries    Examples:

HAP_CongruenceSubgroupGamma0    Examples: 1 

HAP_CongruenceSubgroupGamma0Ideal    Examples:

HAP_ConjugatedCongruenceSubgroup    Examples:

HAP_ConjugatedCongruenceSubgroupGamma0    Examples:

HAP_CriticalCellsDirected    Examples:

HAP_CupProductOfPresentation    Examples:

HAP_CupProductOfSimplicialComplex    Examples:

HAP_DisplayPlanarTree    Examples:

HAP_DisplayVectorField    Examples:

HAP_ElementsSL2Zfn    Examples:

HAP_FunctorialModPCohomologyRing    Examples:

HAP_GenericSL2OSubgroup    Examples:

HAP_GenericSL2ZConjugatedSubgroup    Examples:

HAP_GenericSL2ZSubgroup    Examples:

HAP_HomToIntModP_ChainComplex    Examples:

HAP_HomToIntModP_ChainMap    Examples:

HAP_HomToIntModP_CochainComplex    Examples:

HAP_HomToIntModP_CochainMap    Examples:

HAP_HomeoLinkingForm    Examples:

HAP_Hurewicz1Cycles    Examples:

HAP_IntegralClassToCocycle    Examples:

HAP_IntegralCocycleToClass    Examples:

HAP_IntegralCohomology    Examples:

HAP_KK_AddCell    Examples:

HAP_KnotGroupInv    Examples:

HAP_MyIsBieberbachFpGroup    Examples:

HAP_MyIsFiniteFpGroup    Examples:

HAP_MyIsInfiniteFpGroup    Examples:

HAP_PHI    Examples:

HAP_PermBinlisttoint    Examples:

HAP_PlanarBinaryTrees    Examples:

HAP_PlanarTreeGraft    Examples:

HAP_PlanarTreeJoin    Examples:

HAP_PlanarTreeLeaves    Examples:

HAP_PlanarTreeRemovableEdge    Examples:

HAP_PlanarTreeRemoveEdge    Examples:

HAP_PrimePartModified    Examples:

HAP_PrincipalCongruenceSubgroup    Examples:

HAP_PrincipalCongruenceSubgroupIdeal    Examples:

HAP_PrintTo    Examples:

HAP_PureComplexSubcomplex    Examples:

HAP_PureCubicalPairToCWMap    Examples:

HAP_ResolutionAbelianGroupFromInvariants    Examples:

HAP_RightTransversalSL2ZSubgroups    Examples:

HAP_SL2OSubgroupTree_slow    Examples:

HAP_SL2SubgroupTree    Examples:

HAP_SL2TreeDisplay    Examples:

HAP_SL2ZSubgroupTree_fast    Examples:

HAP_SL2ZSubgroupTree_slow    Examples:

HAP_Sequence2Boundaries    Examples:

HAP_SimplicialPairToCWMap    Examples:

HAP_SimplicialProjectivePlane    Examples:

HAP_SimplicialTorus    Examples:

HAP_SimplifiedGaussCode    Examples:

HAP_StiefelWhitney    Examples:

HAP_SylowSubgroups    Examples:

HAP_Tensor    Examples:

HAP_TransversalCongruenceSubgroups    Examples:

HAP_TransversalCongruenceSubgroupsIdeal    Examples:

HAP_TransversalCongruenceSubgroupsIdeal_alt    Examples:

HAP_TransversalGamma0SubgroupsIdeal    Examples:

HAP_Triangulation    Examples:

HAP_TzPair    Examples:

HAP_WedgeSumOfSimplicialComplexes    Examples:

HAP_bockstein    Examples:

HAP_chain_bockstein    Examples:

HAP_coho_isoms    Examples:

HAP_nxnMatTo2nx2nMat    Examples:

HadamardGraph    Examples:

HapExample    Examples:

HapFile    Examples: 1 , 2 , 3 , 4 

HasTrivialPostnikovInvariant    Examples:

HeckeOperator    Examples:

HeckeOperatorWeight2    Examples:

HenonOrbit    Examples: 1 

HomToGModule_hom    Examples:

HomToInt_ChainComplex    Examples:

HomToInt_ChainMap    Examples:

HomToInt_CochainComplex    Examples:

HomToModPModule    Examples: 1 

HomogeneousPolynomials    Examples:

HomogeneousPolynomials_Bianchi    Examples:

HomologicalGroupDecomposition    Examples: 1 

HomologyOfPureCubicalComplex    Examples:

HomologyPbs    Examples:

HomologySimplicialFreeAbelianGroup    Examples:

HomomorphismAsMatrix    Examples:

HomotopyCatOneGroup    Examples:

HomotopyCrossedModule    Examples:

HomotopyEquivalentLargerSubArray    Examples:

HomotopyEquivalentLargerSubArray3D    Examples:

HomotopyEquivalentLargerSubMatrix    Examples:

HomotopyEquivalentLargerSubPermArray    Examples:

HomotopyEquivalentLargerSubPermArray3D    Examples:

HomotopyEquivalentLargerSubPermMatrix    Examples:

HomotopyEquivalentMaximalPureSubcomplex    Examples:

HomotopyEquivalentMinimalPureSubcomplex    Examples:

HomotopyEquivalentSmallerSubArray    Examples:

HomotopyEquivalentSmallerSubArray3D    Examples:

HomotopyEquivalentSmallerSubMatrix    Examples:

HomotopyEquivalentSmallerSubPermArray    Examples:

HomotopyEquivalentSmallerSubPermArray3D    Examples:

HomotopyEquivalentSmallerSubPermMatrix    Examples:

HomotopyLowerCentralSeries    Examples:

HomotopyLowerCentralSeriesOfCrossedModule    Examples:

HomotopyTruncation    Examples:

HopfSatohSurface    Examples: 1 , 2 

HybridSubdivision    Examples:

IdCatOneGroup    Examples: 1 

IdCrossedModule    Examples:

IdQuasiCatOneGroup    Examples:

IdQuasiCrossedModule    Examples:

IdentifyKnot    Examples: 1 

IdentityAmongRelators    Examples: 1 , 2 

ImageOfGOuterGroupHomomorphism    Examples: 1 , 2 

ImageOfMap    Examples:

InducedSteenrodHomomorphisms    Examples:

IntegerSimplicialComplex    Examples: 1 

IntegralCellularHomology    Examples:

IntegralCohomology    Examples:

IntegralCohomologyOfCochainComplex    Examples:

IntegralHomology    Examples: 1 

IntegralHomologyOfChainComplex    Examples:

IntersectionCWSubcomplex    Examples:

IsClosedManifold    Examples: 1 

IsContractibleCube_higherdims    Examples:

IsCrystSameOrbit    Examples:

IsCrystSufficientLattice    Examples:

IsHadamardMatrix    Examples:

IsIntList    Examples:

IsIsomorphismOfAbelianFpGroups    Examples: 1 

IsMetricMatrix    Examples:

IsPeriodicSpaceGroup    Examples: 1 

IsPureComplex    Examples:

IsPureRegularCWComplex    Examples:

IsRigid    Examples: 1 

IsRigidOnRight    Examples:

IsSphericalCoxeterGroup    Examples:

IsoclinismClasses    Examples: 1 , 2 

IsomorphismCatOneGroups    Examples: 1 

IsomorphismCrossedModules    Examples:

KernelOfGOuterGroupHomomorphism    Examples: 1 , 2 

KernelOfMap    Examples:

KernelWG    Examples:

KinkArc2Presentation    Examples:

KnotComplement    Examples: 1 , 2 , 3 

KnotComplementWithBoundary    Examples: 1 , 2 , 3 

LazyList    Examples:

LefschetzNumberOfChainMap    Examples:

Lfunction    Examples:

LiftColouredSurface    Examples:

LiftedRegularCWMap    Examples:

LinearHomomorphismsZZPersistenceMat    Examples:

LinkingForm    Examples: 1 

LinkingFormHomeomorphismInvariant    Examples: 1 

LinkingFormHomotopyInvariant    Examples: 1 

ListsOfCellsToRegularCWComplex    Examples:

LowDimensionalCupProduct    Examples: 1 

MakeHAPprimeDoc    Examples:

ManifoldType    Examples: 1 

Mapper    Examples: 1 

Mapper_alt    Examples:

MatrixSize    Examples:

MaximalSimplicesOfSimplicialComplex    Examples: 1 

MaximalSphericalCoxeterSubgroupsFromAbove    Examples:

MinimizeRingRelations    Examples:

Mod2SteenrodAlgebra    Examples: 1 

ModPCohomologyRing_alt    Examples:

ModPCohomologyRing_part_1    Examples:

ModPCohomologyRing_part_2    Examples:

ModPRingGeneratorsAlt    Examples:

ModPSteenrodAlgebra    Examples: 1 , 2 

ModularCohomology    Examples:

ModularEquivariantChainMap    Examples:

ModularHomology    Examples:

Nil3TensorSquare    Examples:

NonFreeResolutionFiniteSubgroup    Examples:

NonManifoldVertices    Examples:

NonRegularCWBoundary    Examples:

NonabelianSymmetricKernel_alt    Examples: 1 

NonabelianSymmetricSquare_inf    Examples:

NonabelianTensorProduct_Inf    Examples:

NonabelianTensorProduct_alt    Examples:

NonabelianTensorSquareAsCatOneGroup    Examples:

NonabelianTensorSquareAsCrossedModule    Examples:

NonabelianTensorSquare_inf    Examples:

NoncrossingPartitionsLatticeDisplay    Examples: 1 

NullspaceSparseMatDestructive    Examples:

NumberConnectedQuandles    Examples:

NumberGeneratorsOfGroupHomology    Examples:

NumberOfCrossingsInArc2Presentation    Examples:

NumberOfHomomorphisms_connected    Examples:

NumberOfHomomorphisms_groups    Examples:

NumberOfPrimeKnots    Examples: 1 , 2 

NumberSmallCatOneGroups    Examples:

NumberSmallCrossedModules    Examples:

NumberSmallQuasiCatOneGroups    Examples:

NumberSmallQuasiCrossedModules    Examples:

OppositeGroup    Examples:

OrthogonalizeBasisByAverageInnerProduct    Examples:

PCentre    Examples:

PSubgroupGChainComplex    Examples:

PSubgroupSimplicialComplex    Examples:

PUpperCentralSeries    Examples:

PartialIsoclinismClasses    Examples: 1 

PartsOfQuadraticInteger    Examples:

PathComponentOfPureComplex    Examples: 1 

PathComponentsCWSubcomplex    Examples:

PathComponentsOfSimplicialComplex_alt    Examples:

PathObjectForChainComplex    Examples: 1 

PermutahedralComplexToRegularCWComplex    Examples: 1 

PermutahedralToCubicalArray    Examples:

PersistentBettiNumbersViaContractions    Examples:

PersistentHomologyOfCrossedModule    Examples:

PersistentHomologyOfFilteredPureCubicalComplex_alt    Examples:

PersistentHomologyOfFilteredSparseChainComplex    Examples: 1 , 2 

PersistentHomologyOfPureCubicalComplex_Alt    Examples:

PersistentHomologyOfQuotientGroupSeries_Int    Examples:

PiZeroOfRegularCWComplex    Examples:

PoincareBipyramidCWComplex    Examples: 1 

PoincareCubeCWComplex    Examples: 1 

PoincareCubeCWComplexNS    Examples: 1 

PoincareDodecahedronCWComplex    Examples: 1 

PoincareOctahedronCWComplex    Examples: 1 

PoincarePrismCWComplex    Examples: 1 

PoincareSeriesApproximation    Examples:

PoincareSeries_alt    Examples:

PolymakeFaceLattice    Examples:

PolytopalRepresentationComplex    Examples:

PrankAlt    Examples:

PresentationOfResolution_alt    Examples:

PrimePartDerivedFunctorHomomorphism    Examples:

PrimePartDerivedFunctorViaSubgroupChain    Examples:

PrimePartDerivedTwistedFunctor    Examples:

PrintAlgebraWordAsPolynomial    Examples:

PrintTorsionSubcomplex    Examples:

PureComplex    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 

PureCubicalComplexToCubicalComplex    Examples: 1 , 2 

PureCubicalLink    Examples: 1 , 2 

PushoutOfFpGroups    Examples:

QuadraticCharacter    Examples:

QuadraticNumberField    Examples: 1 

QuandleIsomorphismRepresentatives    Examples:

QuotientByTorsionSubcomplex    Examples:

QuotientChainMap    Examples:

QuotientGroup    Examples:

QuotientQuasiIsomorph    Examples:

RadicalSeriesOfResolution    Examples:

RandomArc2Presentation    Examples:

RandomCellOfPureComplex    Examples:

ReadLinkImageAsGaussCode    Examples: 1 

ReadMatrixAsPureCubicalComplex    Examples:

ReduceGenerators    Examples:

ReduceGenerators_alt    Examples:

ReflectedCubicalKnot    Examples: 1 , 2 , 3 , 4 

RegularCWAssociahedron    Examples:

RegularCWComplexComplement    Examples: 1 

RegularCWComplexWithRemovedCell    Examples: 1 

RegularCWComplex_AttachCellDestructive    Examples: 1 

RegularCWCube    Examples:

RegularCWMapToCWSubcomplex    Examples:

RegularCWOrbitPolytope    Examples:

RegularCWPermutahedron    Examples:

RegularCWPolygon    Examples:

RegularCWSimplex    Examples:

RelativeCentralQuotientSpaceGroup    Examples:

RelativeGroupHomology    Examples:

RelativeRightTransversal    Examples:

RemoveStar    Examples:

ResolutionAbelianGroup_alt    Examples:

ResolutionAbelianPcpGroup    Examples:

ResolutionAffineCrystGroup    Examples:

ResolutionBoundaryOfWordOnRight    Examples:

ResolutionDirectProductLazy    Examples:

ResolutionFiniteCyclicGroup    Examples:

ResolutionGL2QuadraticIntegers    Examples:

ResolutionGL3QuadraticIntegers    Examples:

ResolutionGenericGroup    Examples:

ResolutionInfiniteCyclicGroup    Examples:

ResolutionPGL2QuadraticIntegers    Examples:

ResolutionPGL3QuadraticIntegers    Examples:

ResolutionPSL2QuadraticIntegers    Examples: 1 

ResolutionPrimePowerGroupSparse    Examples:

ResolutionSL2QuadraticIntegers    Examples: 1 

ResolutionSL2ZConjugated    Examples:

ResolutionSL2Z_alt    Examples:

ResolutionSpaceGroup    Examples: 1 

ResolutionToEquivariantCWComplex    Examples:

ResolutionToResolutionOfFpGroup    Examples: 1 

SL2QuadraticIntegers    Examples: 1 

SL2ZResolution    Examples:

SL2ZResolution_alt    Examples:

SL2ZTree    Examples:

SL2ZmElementsDecomposition    Examples:

SequentialRegularCWComplexComplement    Examples:

SignatureOfSymmetricMatrix    Examples: 1 

SignedPermutationGroup    Examples: 1 

SimplicesToSimplicialComplex    Examples: 1 , 2 , 3 , 4 

SimplicialComplexToRegularCWComplex_alt    Examples:

SimplicialK3Surface    Examples: 1 

SimplicialNerveOfFilteredGraph    Examples: 1 , 2 

SimplicialNerveOfTwoComplex    Examples:

SimplifiedQuandlePresentation    Examples:

SimplifiedRegularCWComplex    Examples: 1 

SimplifiedSparseChainComplex    Examples:

SmallCatOneGroup    Examples: 1 

SmallCrossedModule    Examples:

SmallQuasiCatOneGroup    Examples:

SmallQuasiCrossedModule    Examples:

SmoothedFpGroup    Examples:

SparseChainComplexOfCubicalComplex    Examples:

SparseChainComplexOfCubicalPair    Examples:

SparseChainComplexOfFilteredRegularCWComplex    Examples:

SparseChainComplexOfRegularCWComplexWithVectorField    Examples:

SparseChainComplexOfSimplicialComplex    Examples:

SparseChainComplexToChainComplex    Examples:

SparseChainMapOfCubicalPairs    Examples:

SparseFilteredChainComplexOfFilteredCubicalComplex    Examples:

SparseFilteredChainComplexOfFilteredSimplicialComplex    Examples: 1 , 2 

SparseMattoMat    Examples: 1 

SparseRowReduce    Examples:

SphericalKnotComplement    Examples: 1 

Spin    Examples:

SpunAboutHyperplane    Examples:

SpunKnotComplement    Examples: 1 

SpunLinkComplement    Examples:

StrongGeneratorsOfDerivedSubgroup    Examples:

StrongGeneratorsOfDerivedSubgroup_alt    Examples:

StructuralCopyOfFilteredRegularCWComplex    Examples:

SubQuasiIsomorph    Examples:

SubdivideCell    Examples:

Suspension_alt    Examples:

SylowSubgroupOfCatOneGroup    Examples:

SymmetricCentre    Examples:

SymmetricCommutativityGroup    Examples:

TensorNonFreeResolutionWithRationals    Examples:

TensorWithBurnsideRing    Examples: 1 , 2 

TensorWithComplexRepresentationRing    Examples: 1 , 2 

TensorWithComplexRepresentationRingOnRight    Examples:

TensorWithIntegersModPSparse    Examples:

TensorWithIntegersOverSubgroup    Examples: 1 , 2 , 3 , 4 

TensorWithIntegersSparse    Examples:

TensorWithModPModule    Examples: 1 

TestHapBook    Examples:

TestHapQuick    Examples:

ThickenedHEPureCubicalComplex    Examples:

ThickenedPureComplex    Examples: 1 

ThickenedPureCubicalComplex_dim2    Examples:

ThirdHomotopyGroupOfSuspensionB_alt    Examples: 1 

ThreeManifoldViaDehnSurgery    Examples: 1 

ThreeManifoldWithBoundary    Examples: 1 

TransferChainMap    Examples: 1 

TransferCochainMap    Examples: 1 

TranslationSubGroup    Examples:

TreeOfResolutionsToSL2Zcomplex    Examples:

TruncatedRegularCWComplex    Examples:

Tube    Examples:

TupleOrbitReps    Examples:

TupleOrbitReps_perm    Examples:

TwistedResolution    Examples:

UnboundedArrayAssign    Examples:

UnitBall    Examples:

UnitCubicalBall    Examples:

UnitPermutahedralBall    Examples:

UniversalBarCodeEval    Examples:

UniversalCover    Examples: 1 , 2 , 3 , 4 

VectorToCrystMatrix    Examples:

VectorsToOneSkeleton    Examples: 1 

VerticesOfRegularCWCell    Examples:

View3dPureComplex    Examples:

ViewArc2Presentation    Examples:

ViewPureComplex    Examples:

VirtuallySimplicialSubdivision    Examples:

WeakCommutativityGroup    Examples:

WirtingerGroup    Examples: 1 

WirtingerGroup_gc    Examples:

WordModP    Examples:

ZigZagContractedFilteredPureCubicalComplex    Examples:

ZigZagContractedPureComplex    Examples: 1 

Sq    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 

Category_Of_Groups    Examples: 1 

ElementsSL2Z    Examples:

HAP_knot_census    Examples:

PathComponentOfSimplicialComplex    Examples:

ResolutionSL2ZInvertedInteger    Examples:

ViewGraph    Examples:

AsFpGroup    Examples:

BarycentricSubdivision    Examples: 1 , 2 

Bockstein    Examples: 1 , 2 , 3 

CategoryArrow    Examples:

CategoryObject    Examples:

ClosedSurface    Examples: 1 

CoboundaryMatrix    Examples:

CoefficientsOfPoincareSeries    Examples:

CohomologyClass    Examples: 1 , 2 

CohomologyRing    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 

ComplexProjectiveSpace    Examples: 1 

CompositionRingHomomorphism    Examples:

ConnectedComponentsQuandle    Examples:

ConnectedSum    Examples: 1 , 2 

DegreeOfRepresentative    Examples:

Dimensions    Examples:

ExcisedPair    Examples:

FilteredRegularCWComplex    Examples: 1 

FundamentalGroupWithPathReps    Examples: 1 , 2 

GDerivedSubgroup    Examples:

GModuleAsGOuterGroup    Examples: 1 , 2 , 3 , 4 

GOuterGroupHomomorphism    Examples: 1 , 2 

GOuterGroupHomomorphism    Examples: 1 , 2 

GradedAlgebraPresentation    Examples:

GradedAlgebraPresentationNC    Examples:

HAPDerivationNC    Examples:

HAPRingHomomorphismByIndeterminateMap    Examples:

HAPRingReductionHomomorphism    Examples:

HAPRingReductionHomomorphism    Examples:

HAPRingToSubringHomomorphism    Examples:

HAPSubringToRingHomomorphism    Examples:

HAPSubringToRingHomomorphism    Examples:

HAPZeroRingHomomorphism    Examples:

HAP_EquivalenceClasses    Examples:

HomomorphismsImages    Examples:

ImageOfDerivation    Examples:

ImageOfRingHomomorphism    Examples:

IsAssociatedGradedRing    Examples:

KernelOfDerivation    Examples:

LowerGCentralSeries    Examples:

PathComponents    Examples: 1 

PersistentBettiNumbersAlt    Examples: 1 

PersistentHomology    Examples: 1 , 2 

PersistentHomology    Examples: 1 , 2 

PersistentHomology    Examples: 1 , 2 

PoincareSeriesAutoMem    Examples:

PoincareSeriesAutoMem    Examples:

PoincareSeriesAutoMemStop    Examples:

PolynomialToRModuleRep    Examples:

PreimageOfRingHomomorphism    Examples:

PureComplexMeet    Examples:

PureComplexRandomCell    Examples:

PureComplexSubcomplex    Examples:

Pushout    Examples:

QuadraticIdeal    Examples: 1 

ReduceIdeal    Examples:

ReducedPolynomialRingPresentation    Examples:

ReducedPolynomialRingPresentationMap    Examples:

RefinedColouring    Examples:

Resolution    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 

RightTransversal_alt    Examples:

RingOfIntegers    Examples: 1 

SingularPolynomialNormalForm    Examples:

SingularSetNormalFormIdeal    Examples:

SingularSetNormalFormIdealNC    Examples:

SparseChainComplexOfPair    Examples:

Sphere    Examples: 1 

Sq    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 

Standard2Cocycle    Examples:

Standard2Cocycle    Examples:

StandardNCocycle    Examples:

StandardNCocycle    Examples:

StarGraph    Examples:

SubspaceBasisRepsByDegree    Examples:

SubspaceDimensionDegree    Examples:

Suspension    Examples: 1 , 2 , 3 , 4 , 5 

TrivialGModuleAsGOuterGroup    Examples: 1 , 2 , 3 , 4 

VertexLink    Examples:

VertexStar    Examples:

WedgeSum    Examples: 1 

TensorProductOp    Examples:

Arity    Examples:

AssociatedNumberField    Examples:

AssociatedRing    Examples:

Base    Examples: 1 

BaseElement    Examples:

BaseRing    Examples:

Cocycle    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 

CoefficientModule    Examples:

CohomologicalPeriod    Examples: 1 

CoxeterMatrix    Examples: 1 

DerivationImages    Examples:

DerivationRelations    Examples:

DerivationRing    Examples:

Fibre    Examples:

FibreElement    Examples:

Generators    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 

GeneratorsOfPresentationIdeal    Examples:

GradedAlgebraPresentationFamily    Examples:

HAPDerivationFamily    Examples:

HAPPRIME_HilbertSeries    Examples:

HAPRingHomomorphismFamily    Examples:

HAP_MultiplicativeGenerators    Examples:

IdentityMap    Examples:

ImageGenerators    Examples:

ImagePolynomialRing    Examples:

ImageRelations    Examples:

InCcGroup    Examples:

IndeterminateAndExponentOfUnivariateMonomial    Examples:

IndeterminateDegrees    Examples:

IndeterminatesOfGradedAlgebraPresentation    Examples:

IndeterminatesOfPolynomial    Examples:

IndexInSL2O    Examples: 1 

InnerAutomorphismGroupQuandle    Examples:

InnerAutomorphismGroupQuandleAsPerm    Examples:

InverseRingHomomorphism    Examples:

IsConnected    Examples: 1 , 2 , 3 

IsHomogeneousQuandle    Examples:

IsLatinQuandle    Examples: 1 

MaximumDegreeForPresentation    Examples:

ModPRingBasisAsPolynomials    Examples:

ModPRingGeneratorDegrees    Examples:

ModPRingNiceBasis    Examples:

ModPRingNiceBasisAsPolynomials    Examples:

Module    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 

NormOfIdeal    Examples:

OuterAction    Examples:

OuterGroup    Examples: 1 , 2 , 3 , 4 

PresentationIdeal    Examples:

PresentationOfGradedStructureConstantAlgebra    Examples: 1 

Pullbacks    Examples:

Pushouts    Examples:

RightMultiplicationGroupOfQuandle    Examples: 1 , 2 , 3 

RightMultiplicationGroupOfQuandleAsPerm    Examples: 1 

SingularGroebnerBasis    Examples:

SingularReducedGroebnerBasis    Examples:

SourceGenerators    Examples:

SourcePolynomialRing    Examples:

SourceRelations    Examples:

StarGraphAttr    Examples:

TermsOfPolynomial    Examples:

UnivariateMonomialsOfMonomial    Examples:

CoefficientsRing    Examples:

ElementsFamily    Examples:

IndexInSL2Z    Examples:

Name    Examples: 1 , 2 , 3 , 4 , 5 , 6 

Order    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 

IsAbelianCategory    Examples:

IsAdditiveCategory    Examples:

IsCategoryName    Examples:

IsCcGroup    Examples:

IsCrystTranslationSubGroup    Examples:

IsGOuterGroup    Examples:

IsGOuterGroupHomomorphism    Examples:

IsGammaSubgroupInSL3Z    Examples:

IsHAPRationalMatrixGroup    Examples:

IsHAPRationalSpecialLinearGroup    Examples:

IsIdealOfQuadraticIntegers    Examples:

IsPeriodic    Examples: 1 , 2 

IsPseudoListWithFunction    Examples:

IsQuadraticNumberField    Examples:

IsRingOfQuadraticIntegers    Examples:

IsStandard2Cocycle    Examples:

IsStandardNCocycle    Examples:

IsCcElement    Examples:

IsGradedAlgebraPresentation    Examples:

IsHAPDerivation    Examples:

IsHAPRingHomomorphism    Examples:

IsHAPRingModIdealObj    Examples:

IsHapCatOneGroup    Examples:

IsHapCatOneGroupMorphism    Examples:

IsHapChainComplex    Examples:

IsHapChainMap    Examples:

IsHapCochainComplex    Examples:

IsHapCochainMap    Examples:

IsHapCommutativeDiagram    Examples:

IsHapConjQuandElt    Examples:

IsHapCrossedModule    Examples:

IsHapCrossedModuleMorphism    Examples:

IsHapCubicalComplex    Examples:

IsHapEquivariantCWComplex    Examples:

IsHapEquivariantChainComplex    Examples:

IsHapEquivariantChainMap    Examples:

IsHapEquivariantNonFreeChainComplex    Examples:

IsHapEquivariantSpectralSequencePage    Examples:

IsHapFilteredChainComplex    Examples:

IsHapFilteredCubicalComplex    Examples:

IsHapFilteredGraph    Examples:

IsHapFilteredPureCubicalComplex    Examples:

IsHapFilteredRegularCWComplex    Examples:

IsHapFilteredSimplicialComplex    Examples:

IsHapFilteredSparseChainComplex    Examples:

IsHapGCocomplex    Examples:

IsHapGComplex    Examples:

IsHapGComplexMap    Examples:

IsHapGraph    Examples:

IsHapOppositeElement    Examples:

IsHapPureCubicalComplex    Examples:

IsHapPureCubicalLink    Examples:

IsHapPurePermutahedralComplex    Examples:

IsHapQuandlePresentation    Examples:

IsHapQuotientElement    Examples:

IsHapRegularCWComplex    Examples:

IsHapRegularCWMap    Examples:

IsHapResolution    Examples:

IsHapSimplicialComplex    Examples:

IsHapSimplicialFreeAbelianGroup    Examples:

IsHapSimplicialGroup    Examples:

IsHapSimplicialGroupMorphism    Examples:

IsHapSimplicialMap    Examples:

IsHapSparseChainComplex    Examples:

IsHapSparseChainMap    Examples:

IsHapSparseMat    Examples:

IsHapTorsionSubcomplex    Examples:

IsPseudoList    Examples:

IsCcElementRep    Examples:

IsGradedAlgebraPresentationRep    Examples:

IsHAPDerivationRep    Examples:

IsHAPIdealRep    Examples:

IsHAPRingHomomorphismIndeterminateMapRep    Examples:

IsHAPRingReductionHomomorphismRep    Examples:

IsHAPRingToSubringHomomorphismRep    Examples:

IsHAPSubringToRingHomomorphismRep    Examples:

IsHAPZeroRingHomomorphismRep    Examples:

IsHapCatOneGroupMorphismRep    Examples:

IsHapCatOneGroupRep    Examples:

IsHapChainComplexRep    Examples:

IsHapChainMapRep    Examples:

IsHapCochainComplexRep    Examples:

IsHapCochainMapRep    Examples:

IsHapCommutativeDiagramRep    Examples:

IsHapConjQuandEltRep    Examples:

IsHapCrossedModuleMorphismRep    Examples:

IsHapCrossedModuleRep    Examples:

IsHapCubicalComplexRep    Examples:

IsHapEquivariantCWComplexRep    Examples:

IsHapEquivariantChainComplexRep    Examples:

IsHapEquivariantChainMapRep    Examples:

IsHapEquivariantNonFreeChainComplexRep    Examples:

IsHapEquivariantSpectralSequencePageRep    Examples:

IsHapFilteredChainComplexRep    Examples:

IsHapFilteredCubicalComplexRep    Examples:

IsHapFilteredGraphRep    Examples:

IsHapFilteredPureCubicalComplexRep    Examples:

IsHapFilteredRegularCWComplexRep    Examples:

IsHapFilteredSimplicialComplexRep    Examples:

IsHapFilteredSparseChainComplexRep    Examples:

IsHapGCocomplexRep    Examples:

IsHapGComplexMapRep    Examples:

IsHapGComplexRep    Examples:

IsHapGraphRep    Examples:

IsHapOppositeElementRep    Examples:

IsHapPureCubicalComplexRep    Examples:

IsHapPureCubicalLinkRep    Examples:

IsHapPurePermutahedralComplexRep    Examples:

IsHapQuandlePresentationRep    Examples:

IsHapQuotientElementRep    Examples:

IsHapRegularCWComplexRep    Examples:

IsHapRegularCWMapRep    Examples:

IsHapResolutionRep    Examples:

IsHapSimplicialComplexRep    Examples:

IsHapSimplicialFreeAbelianGroupRep    Examples:

IsHapSimplicialGroupMorphismRep    Examples:

IsHapSimplicialGroupRep    Examples:

IsHapSimplicialMapRep    Examples:

IsHapSparseChainComplexRep    Examples:

IsHapSparseChainMapRep    Examples:

IsHapSparseMatRep    Examples:

IsHapTorsionSubcomplexRep    Examples:

IsPseudoListRep    Examples:

IdealOfQuadraticIntegers    Examples:

QuadraticNF    Examples:

RingOfQuadraticIntegers    Examples:

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

+    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 

+    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 

+    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 

    Examples:

    Examples:

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

AdditiveInverseMutable    Examples:

AsFpGroup    Examples:

AsList    Examples:

AsSSortedList    Examples:

BarycentricSubdivision    Examples: 1 , 2 

BaseRing    Examples:

Bockstein    Examples: 1 , 2 , 3 

CategoryArrow    Examples:

CategoryObject    Examples:

CoboundaryMatrix    Examples:

CoefficientsOfPoincareSeries    Examples:

CoefficientsRing    Examples:

CohomologicalPeriod    Examples: 1 

CohomologyClass    Examples: 1 , 2 

CompositionRingHomomorphism    Examples:

CompositionRingHomomorphism    Examples:

CompositionRingHomomorphism    Examples:

CompositionRingHomomorphism    Examples:

CompositionRingHomomorphism    Examples:

ConnectedComponentsQuandle    Examples:

ConnectedSum    Examples: 1 , 2 

CoxeterMatrix    Examples: 1 

DefaultFieldOfMatrixGroup    Examples:

DegreeOfRepresentative    Examples:

DerivationImages    Examples:

DerivationRelations    Examples:

DerivationRing    Examples:

Dimensions    Examples:

Enumerator    Examples:

ExcisedPair    Examples:

FilteredRegularCWComplex    Examples: 1 

FundamentalGroupWithPathReps    Examples: 1 , 2 

GModuleAsGOuterGroup    Examples: 1 , 2 , 3 , 4 

GOuterGroupHomomorphism    Examples: 1 , 2 

GOuterGroupHomomorphism    Examples: 1 , 2 

Generators    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 

GeneratorsOfMagmaWithInverses    Examples:

GeneratorsOfMagmaWithInverses    Examples:

GeneratorsOfPresentationIdeal    Examples:

GradedAlgebraPresentation    Examples:

GradedAlgebraPresentationNC    Examples:

HAPDerivationNC    Examples:

HAPPRIME_HilbertSeries    Examples:

HAPRingHomomorphismByIndeterminateMap    Examples:

HAPRingReductionHomomorphism    Examples:

HAPRingReductionHomomorphism    Examples:

HAPRingToSubringHomomorphism    Examples:

HAPSubringToRingHomomorphism    Examples:

HAPSubringToRingHomomorphism    Examples:

HAPZeroRingHomomorphism    Examples:

HAP_MultiplicativeGenerators    Examples:

HomomorphismsImages    Examples:

IdGroup    Examples: 1 , 2 , 3 , 4 , 5 , 6 

IdentityMap    Examples:

ImageOfDerivation    Examples:

ImageOfRingHomomorphism    Examples:

ImageOfRingHomomorphism    Examples:

ImageOfRingHomomorphism    Examples:

ImageOfRingHomomorphism    Examples:

ImageOfRingHomomorphism    Examples:

IndeterminateAndExponentOfUnivariateMonomial    Examples:

IndeterminateDegrees    Examples:

IndeterminatesOfGradedAlgebraPresentation    Examples:

IndeterminatesOfPolynomial    Examples:

IndexInSL2O    Examples: 1 

IndexInSL2Z    Examples:

InnerAutomorphismGroupQuandle    Examples:

InnerAutomorphismGroupQuandleAsPerm    Examples:

Int    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 

InverseMutable    Examples:

InverseMutable    Examples:

InverseMutable    Examples:

InverseRingHomomorphism    Examples:

InverseRingHomomorphism    Examples:

InverseRingHomomorphism    Examples:

InverseRingHomomorphism    Examples:

InverseRingHomomorphism    Examples:

InverseSameMutability    Examples:

IsAssociatedGradedRing    Examples:

IsConnected    Examples: 1 , 2 , 3 

IsHomogeneousQuandle    Examples:

IsLatinQuandle    Examples: 1 

IsMonomial    Examples:

IsOne    Examples:

IsPeriodic    Examples: 1 , 2 

Kernel    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 

KernelOfDerivation    Examples:

MaximumDegreeForPresentation    Examples:

ModPRingBasisAsPolynomials    Examples:

ModPRingGeneratorDegrees    Examples:

ModPRingNiceBasis    Examples:

ModPRingNiceBasisAsPolynomials    Examples:

OneImmutable    Examples:

OneMutable    Examples:

PathComponents    Examples: 1 

PersistentBettiNumbersAlt    Examples: 1 

PreimageOfRingHomomorphism    Examples:

PresentationIdeal    Examples:

PresentationOfGradedStructureConstantAlgebra    Examples: 1 

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

Projection    Examples:

PureComplexMeet    Examples:

PureComplexRandomCell    Examples:

PureComplexSubcomplex    Examples:

Pushout    Examples:

QuadraticIdeal    Examples: 1 

Random    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 

ReduceIdeal    Examples:

ReducedPolynomialRingPresentation    Examples:

ReducedPolynomialRingPresentationMap    Examples:

RefinedColouring    Examples:

Resolution    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 

RightMultiplicationGroupOfQuandle    Examples: 1 , 2 , 3 

RightMultiplicationGroupOfQuandleAsPerm    Examples: 1 

RightTransversal    Examples:

RingOfIntegers    Examples: 1 

SingularGroebnerBasis    Examples:

SingularPolynomialNormalForm    Examples:

SingularReducedGroebnerBasis    Examples:

SingularSetNormalFormIdeal    Examples:

SingularSetNormalFormIdealNC    Examples:

SparseChainComplexOfPair    Examples:

Sq    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 

Standard2Cocycle    Examples:

Standard2Cocycle    Examples:

StandardNCocycle    Examples:

StandardNCocycle    Examples:

StarGraph    Examples:

StarGraphAttr    Examples:

SubspaceBasisRepsByDegree    Examples:

SubspaceDimensionDegree    Examples:

Suspension    Examples: 1 , 2 , 3 , 4 , 5 

TensorProductOp    Examples:

TermsOfPolynomial    Examples:

TrivialGModuleAsGOuterGroup    Examples: 1 , 2 , 3 , 4 

Units    Examples:

Units    Examples:

UnivariateMonomialsOfMonomial    Examples:

VertexLink    Examples:

VertexStar    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

WedgeSum    Examples: 1 

ZeroMutable    Examples:

^    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 

^    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 

^    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

mod    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

+    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 

+    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 

+    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 

-    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

-    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

/    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

    Examples:

    Examples:

    Examples:

    Examples:

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

AbelianInvariants    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 

AbelianInvariants    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 

BarycentricSubdivision    Examples: 1 , 2 

Bockstein    Examples: 1 , 2 , 3 

CanonicalRightCosetElement    Examples:

ClosedSurface    Examples: 1 

CohomologyRing    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 

CohomologyRing    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 

ComplexProjectiveSpace    Examples: 1 

ConnectedSum    Examples: 1 , 2 

ConnectedSum    Examples: 1 , 2 

ConnectedSum    Examples: 1 , 2 

Dimensions    Examples:

DirectProductOp    Examples:

DirectProductOp    Examples:

DirectProductOp    Examples:

DirectProductOp    Examples:

Discriminant    Examples:

Discriminant    Examples:

Embedding    Examples:

GDerivedSubgroup    Examples:

Generators    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 

HAPRingReductionHomomorphism    Examples:

HAPRingReductionHomomorphism    Examples:

HAP_EquivalenceClasses    Examples:

ImageOfRingHomomorphism    Examples:

ImageOfRingHomomorphism    Examples:

IndexNC    Examples:

IndexNC    Examples:

InverseMutable    Examples:

InverseMutable    Examples:

InverseMutable    Examples:

InverseSameMutability    Examples:

IsEmpty    Examples:

IsEmpty    Examples:

IsPrime    Examples: 1 , 2 

IsomorphismFpGroup    Examples: 1 , 2 

Iterator    Examples:

KernelOfDerivation    Examples:

ListOp    Examples:

ListOp    Examples:

LowerGCentralSeries    Examples:

NaturalHomomorphism    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 

Norm    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 

Norm    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 

OneImmutable    Examples:

OneImmutable    Examples:

Order    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 

Order    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 

PersistentBettiNumbersAlt    Examples: 1 

PersistentBettiNumbersAlt    Examples: 1 

PersistentBettiNumbersAlt    Examples: 1 

PersistentBettiNumbersAlt    Examples: 1 

PersistentBettiNumbersAlt    Examples: 1 

PersistentHomology    Examples: 1 , 2 

PersistentHomology    Examples: 1 , 2 

Position    Examples: 1 , 2 

Position    Examples: 1 , 2 

Position    Examples: 1 , 2 

PositionCanonical    Examples:

PreimageOfRingHomomorphism    Examples:

Projection    Examples:

PureComplexMeet    Examples:

PureComplexRandomCell    Examples:

PureComplexSubcomplex    Examples:

QuadraticIdeal    Examples: 1 

Range    Examples: 1 , 2 

RankMatrixDestructive    Examples:

ReduceIdeal    Examples:

ReduceIdeal    Examples:

ReducedPolynomialRingPresentation    Examples:

ReducedPolynomialRingPresentation    Examples:

ReducedPolynomialRingPresentationMap    Examples:

ReducedPolynomialRingPresentationMap    Examples:

ReducedPolynomialRingPresentationMap    Examples:

RefinedColouring    Examples:

RightTransversal    Examples:

RightTransversal    Examples:

RightTransversal    Examples:

RightTransversal    Examples:

RightTransversal    Examples:

RightTransversal_alt    Examples:

SingularPolynomialNormalForm    Examples:

SparseChainComplexOfPair    Examples:

Sphere    Examples: 1 

SubspaceBasisRepsByDegree    Examples:

SubspaceDimensionDegree    Examples:

Suspension    Examples: 1 , 2 , 3 , 4 , 5 

TensorProductOp    Examples:

TensorProductOp    Examples:

TensorProductOp    Examples:

Trace    Examples:

Units    Examples:

WedgeSum    Examples: 1 

WedgeSum    Examples: 1 

WedgeSum    Examples: 1 

[]    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 

[]    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 

[]    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 

^    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 

^    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 

^    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 

^    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

mod    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 

InfoHAPprime    Examples:

ASY_PATH    Examples:

AutomorphismGroupAsCrossedModule    Examples:

BROWSER_PATH    Examples:

CATONEGROUP_DATA_PERM    Examples:

CATONEGROUP_DATA_SIZE    Examples:

Cedric_PlanarDiagram    Examples:

ChildKill    Examples:

DISPLAY_PATH    Examples:

DOT_PATH    Examples:

FilteredSimplicialComplexToFilteredCWComplex    Examples:

GradedAlgebraPresentationType    Examples:

HAPTEMPORARYFUNCTION    Examples:

HAP_Knots    Examples:

HAP_ROOT    Examples:

HapCatOneGroup    Examples:

HapCatOneGroupFamily    Examples:

HapCatOneGroupMorphism    Examples:

HapCatOneGroupMorphismFamily    Examples:

HapChainComplex    Examples:

HapChainComplexFamily    Examples:

HapChainMap    Examples:

HapChainMapFamily    Examples:

HapCochainComplex    Examples:

HapCochainComplexFamily    Examples:

HapCochainMap    Examples:

HapCochainMapFamily    Examples:

HapCommutativeDiagram    Examples:

HapCommutativeDiagramFamily    Examples:

HapCrossedModule    Examples:

HapCrossedModuleFamily    Examples:

HapCrossedModuleMorphism    Examples:

HapCrossedModuleMorphismFamily    Examples:

HapCubicalComplex    Examples:

HapCubicalComplexFamily    Examples:

HapEquivariantCWComplex    Examples:

HapEquivariantCWComplexFamily    Examples:

HapEquivariantChainMap    Examples:

HapEquivariantChainMapFamily    Examples:

HapFPGModule    Examples:

HapFPGModuleHomomorphism    Examples:

HapFilteredChainComplex    Examples:

HapFilteredChainComplexFamily    Examples:

HapFilteredCubicalComplex    Examples:

HapFilteredCubicalComplexFamily    Examples:

HapFilteredGraph    Examples:

HapFilteredGraphFamily    Examples:

HapFilteredPureCubicalComplex    Examples:

HapFilteredPureCubicalComplexFamily    Examples:

HapFilteredRegularCWComplex    Examples:

HapFilteredRegularCWComplexFamily    Examples:

HapFilteredSimplicialComplex    Examples:

HapFilteredSimplicialComplexFamily    Examples:

HapFilteredSparseChainComplex    Examples:

HapFilteredSparseChainComplexFamily    Examples:

HapGChainComplex    Examples:

HapGCocomplex    Examples:

HapGCocomplexFamily    Examples:

HapGComplex    Examples:

HapGComplexFamily    Examples:

HapGlobalDeclarationsAreAlreadyLoaded    Examples:

HapGraph    Examples:

HapGraphFamily    Examples:

HapNonFreeResolution    Examples:

HapOppositeElement    Examples:

HapOppositeElementFamily    Examples:

HapPureCubicalComplex    Examples:

HapPureCubicalComplexFamily    Examples:

HapPureCubicalLink    Examples:

HapPureCubicalLinkFamily    Examples:

HapPurePermutahedralComplex    Examples:

HapPurePermutahedralComplexFamily    Examples:

HapQuotientElement    Examples:

HapQuotientElementFamily    Examples:

HapRegularCWComplex    Examples:

HapRegularCWComplexFamily    Examples:

HapRegularCWMap    Examples:

HapRegularCWMapFamily    Examples:

HapResolution    Examples:

HapResolutionFamily    Examples:

HapSimplicialComplex    Examples:

HapSimplicialComplexFamily    Examples:

HapSimplicialGroup    Examples:

HapSimplicialGroupFamily    Examples:

HapSimplicialGroupMorphism    Examples:

HapSimplicialGroupMorphismFamily    Examples:

HapSimplicialMap    Examples:

HapSimplicialMapFamily    Examples:

HapSparseChainComplex    Examples:

HapSparseChainComplexFamily    Examples:

HapSparseChainMap    Examples:

HapSparseChainMapFamily    Examples:

HapSparseMat    Examples:

HapSparseMatFamily    Examples:

HomomorphismOfDirectProduct    Examples:

IDQUASICATONEGROUP_DATA    Examples:

IsHapChain    Examples:

IsHapCochain    Examples:

IsHapComplex    Examples:

IsHapFPGModule    Examples:

IsHapFPGModuleHomomorphism    Examples:

IsHapGChainComplex    Examples:

IsHapMap    Examples:

IsHapNonFreeResolution    Examples:

NEATO_PATH    Examples:

NerveOfCover    Examples:

POLYMAKE_PATH    Examples:

PseudoList    Examples:

PseudoListFamily    Examples:

QUASICATONEGROUP_DATA_NOT    Examples:

QUASICATONEGROUP_DATA_SIZE    Examples:

ReadBioData    Examples:

SMALLQUASICATONEGROUP_DATA    Examples:

CATONEGROUP_DATA    Examples:

COMPILED    Examples:

Cedric_XYXYConnQuan    Examples:

Cedric_XYXYQuandles    Examples:

CommutingProbability    Examples:

GroupIsomorphismRepresentatives    Examples:

HAPAAA    Examples:

HAPBARCODE    Examples:

HAPDerivationType    Examples:

HAPPRIME_LastLHSBicomplexSize    Examples:

HAPPRIME_ShuffleRandomSource    Examples:

HAPRIGXXX    Examples:

HAP_GCOMPLEX_LIST    Examples:

HAP_GCOMPLEX_SETUP    Examples:

HAP_MOVES_DIM_2    Examples:

HAP_MOVES_DIM_3    Examples:

HAP_PERMMOVES_DIM_2    Examples:

HAP_PERMMOVES_DIM_3    Examples:

HAP_PoincareCubeManifoldEdgeDegrees    Examples:

HAP_Test    Examples:

HAP_XYXYXYXY    Examples:

HAPchildFunctionToggle    Examples:

HAPchildToggle    Examples:

HAPchildren    Examples:

HapConjQuandElt    Examples:

HapConjQuandEltFamily    Examples:

HapConstantPolRing    Examples:

HapEquivariantChainComplex    Examples:

HapEquivariantChainComplexFamily    Examples:

HapEquivariantNonFreeChainComplex    Examples:

HapEquivariantNonFreeChainComplexFamily    Examples:

HapEquivariantSpectralSequencePage    Examples:

HapEquivariantSpectralSequencePageFamily    Examples:

HapGComplexMap    Examples:

HapGComplexMapFamily    Examples:

HapQuandlePresentation    Examples:

HapQuandlePresentationFamily    Examples:

HapRightTransversalSL2ZSubgroup    Examples:

HapSL2ZConjugatedSubgroup    Examples:

HapSL2ZSubgroup    Examples:

HapSimplicialFreeAbelianGroup    Examples:

HapSimplicialFreeAbelianGroupFamily    Examples:

HapTorsionSubcomplex    Examples:

HapTorsionSubcomplexFamily    Examples:

IntersectionForm    Examples: 1 , 2 

IsHapRightTransversalSL2ZSubgroup    Examples:

IsHapSL2ConjugatedSubgroup    Examples:

IsHapSL2OSubgroup    Examples:

IsHapSL2Subgroup    Examples:

IsHapSL2ZConjugatedSubgroup    Examples:

IsHapSL2ZSubgroup    Examples:

RefinedColouring_gc    Examples:

RefinedColouring_group    Examples:

RegularCWAssociahedronWithDiscreteVectorField    Examples:

RegularCWClosedSurface    Examples:

RegularCWComplexWithAttachedRelatorCells    Examples: 1 

RegularCWComplex_DisjointUnion    Examples:

RegularCWComplex_WedgeSum    Examples:

RegularCWDiscreteSpace    Examples: 1 

RegularCWSphere    Examples: 1 

SimplicialComplexConnectedSum    Examples:

SphericalKnotComplementWithBoundary    Examples:

StemGroups    Examples:

cat    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 

cnt    Examples:

hap_cr    Examples:

+

2CoreducedChainComplex    Examples:

AbelianGOuterGroupToCatOneGroup    Examples:

AbelianInvariantsToTorsionCoefficients    Examples:

AcyclicSubcomplexOfPureCubicalComplex    Examples: 1 

AddFirst    Examples:

AdjointGroupOfQuandle    Examples: 1 

AlgebraicReduction_alt    Examples:

AppendFreeWord    Examples:

ArcDiagramToTubularSurface    Examples:

ArcPresentation    Examples: 1 , 2 , 3 , 4 

ArcPresentationToKnottedOneComplex    Examples:

AreIsoclinic    Examples:

ArrayIterateBreak    Examples:

ArrayValueKD    Examples:

AsWordInSL2Z    Examples:

AutomorphismGroupQuandleAsPerm_nonconnected    Examples:

AverageInnerProduct    Examples:

BarCodeOfFilteredPureCubicalComplex    Examples:

BarCodeOfSymmetricMatrix    Examples:

BarComplexOfMonoid    Examples: 1 

BarycentricallySimplifiedComplex    Examples: 1 

BarycentricallySubdivideCell    Examples:

BettinumbersOfPureCubicalComplex_dim_2    Examples:

BocksteinHomology    Examples:

BogomolovMultiplier_viaTensorSquare    Examples:

BoundariesOfFilteredChainComplex    Examples:

BoundaryOfPureComplex    Examples: 1 

BoundaryOfPureRegularCWComplex    Examples: 1 

BoundaryOfRegularCWCell    Examples:

BoundaryPairOfPureRegularCWComplex    Examples:

BoundingPureComplex    Examples:

CR_ChainMapFromCocycle    Examples:

CR_CocyclesAndCoboundaries    Examples:

CR_IntegralClassToCocycle    Examples:

CR_IntegralCocycleToClass    Examples:

CR_IntegralCohomology    Examples:

CR_IntegralCycleToClass    Examples:

CWMap2ChainMap    Examples:

CWSubcomplexToRegularCWMap    Examples: 1 

CanonicalRightCountableCosetElement    Examples:

CatOneGroupByCrossedModule    Examples:

CatOneGroupsByGroup    Examples:

CcElement    Examples:

Cedric_CheckThirdAxiomRow    Examples:

Cedric_ConjugateQuandleElement    Examples:

Cedric_FromAutGeReToAutQe    Examples:

Cedric_IsHomomorphism    Examples:

Cedric_Permute    Examples:

Cedric_Quandle1    Examples:

Cedric_Quandle2    Examples:

Cedric_Quandle3    Examples:

Cedric_Quandle4    Examples:

Cedric_Quandle5    Examples:

Cedric_Quandle6    Examples:

CellComplexBoundaryCheck    Examples:

ChainComplexEquivalenceOfRegularCWComplex    Examples: 1 

ChainComplexHomeomorphismEquivalenceOfRegularCWComplex    Examples:

ChainComplexOfCubicalComplex    Examples:

ChainComplexOfCubicalPair    Examples:

ChainComplexOfRegularCWComplexWithVectorField    Examples:

ChainComplexOfSimplicialComplex    Examples:

ChainComplexOfSimplicialPair    Examples:

ChainComplexOfUniversalCover    Examples: 1 , 2 , 3 , 4 

ChainComplexToSparseChainComplex    Examples:

ChainComplexWithChainHomotopy    Examples:

ChainMapOfCubicalPairs    Examples:

ChainMapOfRegularCWMap    Examples:

ChevalleyEilenbergComplexOfModule    Examples:

ChildRestart    Examples:

ClosureCWCell    Examples:

CoClass    Examples:

CocriticalCellsOfRegularCWComplex    Examples:

CocyclicHadamardMatrices    Examples: 1 

CocyclicMatrices    Examples:

CohomologicalData    Examples: 1 

CohomologyHomomorphism    Examples: 1 , 2 

CohomologyHomomorphismOfRepresentation    Examples:

CohomologyModule_AsAutModule    Examples:

CohomologyModule_Gmap    Examples:

CohomologyRingOfSimplicialComplex    Examples:

CohomologySimplicialFreeAbelianGroup    Examples:

CombinationDisjointSets    Examples:

CommonEndomorphisms    Examples:

ComplementOfPureComplex    Examples: 1 

ComplementaryBasis    Examples:

ComposeCWMaps    Examples:

CompositionOfFpGModuleHomomorphisms    Examples:

CompositionSeriesOfFpGModule    Examples:

ConcentricallyFilteredPureCubicalComplex    Examples: 1 

CongruenceSubgroup    Examples: 1 , 2 

ConjugateSL2ZGroup    Examples:

ConnectingCohomologyHomomorphism    Examples: 1 , 2 

ContractArray    Examples:

ContractCubicalComplex_dim2    Examples:

ContractCubicalComplex_dim3    Examples:

ContractMatrix    Examples:

ContractPermArray    Examples:

ContractPermMatrix    Examples:

ContractPureComplex    Examples:

ContractSimplicialComplex    Examples:

ContractSimplicialComplex_alt    Examples:

ContractedFilteredPureCubicalComplex    Examples: 1 

ContractedFilteredRegularCWComplex    Examples:

ContractedRegularCWComplex    Examples:

ContractibleSL2ZComplex    Examples:

ContractibleSL2ZComplex_alt    Examples:

ContractibleSubArray    Examples:

ContractibleSubMatrix    Examples:

ContractibleSubcomplexOfPureCubicalComplex    Examples: 1 

ConvertTorsionComplexToGcomplex    Examples:

CosetsQuandle    Examples:

CountingCellsOfBaryCentricSubdivision    Examples:

CountingNumberOfCellsInBaryCentricSubdivision    Examples:

CoxeterComplex_alt    Examples: 1 

CoxeterDiagramMatCoxeterGroup    Examples:

CoxeterWythoffComplex    Examples:

CreateCoxeterMatrix    Examples: 1 

CriticalBoundaryCells    Examples: 1 

CropPureComplex    Examples:

CrossedInvariant    Examples:

CrossedModuleByAutomorphismGroup    Examples:

CrossedModuleByCatOneGroup    Examples:

CrossedModuleByNormalSubgroup    Examples: 1 

CrystCubicalTiling    Examples:

CrystFinitePartOfMatrix    Examples:

CrystGFullBasis    Examples: 1 , 2 

CrystGcomplex    Examples: 1 , 2 

CrystMatrix    Examples:

CrystTranslationMatrixToVector    Examples:

CrystallographicComplex    Examples:

CubicalToPermutahedralArray    Examples:

CupProductMatrix    Examples:

CupProductOfRegularCWComplex    Examples: 1 

CupProductOfRegularCWComplex_alt    Examples: 1 

CuspidalCohomologyHomomorphism    Examples:

CyclesOfFilteredChainComplex    Examples:

DavisComplex    Examples: 1 , 2 , 3 , 4 

DeformationRetract    Examples:

DensityMat    Examples:

DerivedGroupOfQuandle    Examples: 1 

DiagonalChainMap    Examples:

DijkgraafWittenInvariant    Examples: 1 

DirectProductOfGroupHomomorphisms    Examples:

DirectProductOfRegularCWComplexes    Examples:

DirectProductOfRegularCWComplexesLazy    Examples:

DirectProductOfSimplicialComplexes    Examples:

DisplayCSVknotFile    Examples:

DisplayVectorField    Examples:

E1CohomologyPage    Examples:

E1HomologyPage    Examples:

EilenbergMacLaneSimplicialFreeAbelianGroup    Examples:

ElementsLazy    Examples:

EquivariantCWComplexToRegularCWComplex    Examples: 1 , 2 , 3 , 4 

EquivariantCWComplexToRegularCWMap    Examples: 1 , 2 , 3 

EquivariantCWComplexToResolution    Examples:

ExcisedPureCubicalPair_dim_2    Examples:

ExtractTorsionSubcomplex    Examples:

FactorizationNParts    Examples:

FilteredChainComplexToFilteredSparseChainComplex    Examples:

FilteredCubicalComplexToFilteredRegularCWComplex    Examples: 1 

FilteredPureCubicalComplexToCubicalComplex    Examples: 1 

FiltrationTermOfGraph    Examples:

FiltrationTermOfPureCubicalComplex    Examples:

FiltrationTermOfRegularCWComplex    Examples:

FiltrationTerms    Examples: 1 

FirstHomologyCoveringCokernels    Examples: 1 , 2 

FirstHomologySimplicialTwoComplex    Examples:

FourthHomotopyGroupOfDoubleSuspensionB    Examples:

Fp2PcpAbelianGroupHomomorphism    Examples:

FpGModuleSection    Examples:

FreeZGResolution    Examples:

FundamentalGroupOfRegularCWComplex    Examples: 1 

FundamentalGroupOfRegularCWMap    Examples:

FundamentalGroupSimplicialTwoComplex    Examples:

FundamentalMultiplesOfStiefelWhitneyClasses    Examples:

GChainComplex    Examples: 1 

GModuleAsCatOneGroup    Examples:

GammaSubgroupInSL3Z    Examples:

GaussCodeOfPureCubicalKnot    Examples: 1 , 2 , 3 , 4 

GetTorsionPowerSubcomplex    Examples:

GetTorsionSubcomplex    Examples:

GraphOfRegularCWComplex    Examples:

GraphOfResolutionsTest    Examples:

GraphOfResolutionsToGroups    Examples:

GroupHomomorphismToMatrix    Examples:

HAPCocontractRegularCWComplex    Examples:

HAPContractFilteredRegularCWComplex    Examples:

HAPContractRegularCWComplex    Examples:

HAPContractRegularCWComplex_Alt    Examples:

HAPPRIME_Algebra2Polynomial    Examples:

HAPPRIME_CohomologyRingWithoutResolution    Examples:

HAPPRIME_CombineIndeterminateMaps    Examples:

HAPPRIME_GradedAlgebraPresentationAvoidingIndeterminates    Examples:

HAPPRIME_LHSSpectralSequence    Examples:

HAPPRIME_MakeEliminationOrdering    Examples:

HAPPRIME_MapPolynomialIndeterminates    Examples:

HAPPRIME_Polynomial2Algebra    Examples:

HAPPRIME_RingHomomorphismsAreComposable    Examples:

HAPPRIME_SModule    Examples:

HAPPRIME_SingularGroebnerBasis    Examples:

HAPPRIME_SingularReducedGroebnerBasis    Examples:

HAPPRIME_SwitchGradedAlgebraRing    Examples:

HAPPRIME_SwitchPolynomialIndeterminates    Examples:

HAPPRIME_VersionWithSVN    Examples:

HAPRegularCWComplex    Examples:

HAPRegularCWPolytope    Examples:

HAPRemoveCellFromRegularCWComplex    Examples:

HAPRemoveVectorField    Examples:

HAPRingModIdeal    Examples:

HAPRingModIdealObj    Examples:

HAPTietzeReduction_Inf    Examples:

HAPTietzeReduction_OneLevel    Examples:

HAPTietzeReduction_OneStep    Examples:

HAP_4x4MatTo2x2Mat    Examples:

HAP_AddGenerator    Examples:

HAP_AllHomomorphisms    Examples:

HAP_AppendTo    Examples:

HAP_AssociahedronBoundaries    Examples:

HAP_AssociahedronCells    Examples:

HAP_BaryCentricSubdivisionGComplex    Examples:

HAP_BaryCentricSubdivisionRegularCWComplex    Examples:

HAP_Binlisttoint    Examples:

HAP_ChainComplexToEquivariantChainComplex    Examples:

HAP_CocyclesAndCoboundaries    Examples:

HAP_CongruenceSubgroupGamma0    Examples: 1 

HAP_CongruenceSubgroupGamma0Ideal    Examples:

HAP_ConjugatedCongruenceSubgroup    Examples:

HAP_ConjugatedCongruenceSubgroupGamma0    Examples:

HAP_CriticalCellsDirected    Examples:

HAP_CupProductOfPresentation    Examples:

HAP_CupProductOfSimplicialComplex    Examples:

HAP_DisplayPlanarTree    Examples:

HAP_DisplayVectorField    Examples:

HAP_ElementsSL2Zfn    Examples:

HAP_FunctorialModPCohomologyRing    Examples:

HAP_GenericSL2OSubgroup    Examples:

HAP_GenericSL2ZConjugatedSubgroup    Examples:

HAP_GenericSL2ZSubgroup    Examples:

HAP_HomToIntModP_ChainComplex    Examples:

HAP_HomToIntModP_ChainMap    Examples:

HAP_HomToIntModP_CochainComplex    Examples:

HAP_HomToIntModP_CochainMap    Examples:

HAP_HomeoLinkingForm    Examples:

HAP_Hurewicz1Cycles    Examples:

HAP_IntegralClassToCocycle    Examples:

HAP_IntegralCocycleToClass    Examples:

HAP_IntegralCohomology    Examples:

HAP_KK_AddCell    Examples:

HAP_KnotGroupInv    Examples:

HAP_MyIsBieberbachFpGroup    Examples:

HAP_MyIsFiniteFpGroup    Examples:

HAP_MyIsInfiniteFpGroup    Examples:

HAP_PHI    Examples:

HAP_PermBinlisttoint    Examples:

HAP_PlanarBinaryTrees    Examples:

HAP_PlanarTreeGraft    Examples:

HAP_PlanarTreeJoin    Examples:

HAP_PlanarTreeLeaves    Examples:

HAP_PlanarTreeRemovableEdge    Examples:

HAP_PlanarTreeRemoveEdge    Examples:

HAP_PrimePartModified    Examples:

HAP_PrincipalCongruenceSubgroup    Examples:

HAP_PrincipalCongruenceSubgroupIdeal    Examples:

HAP_PrintTo    Examples:

HAP_PureComplexSubcomplex    Examples:

HAP_PureCubicalPairToCWMap    Examples:

HAP_ResolutionAbelianGroupFromInvariants    Examples:

HAP_RightTransversalSL2ZSubgroups    Examples:

HAP_SL2OSubgroupTree_slow    Examples:

HAP_SL2SubgroupTree    Examples:

HAP_SL2TreeDisplay    Examples:

HAP_SL2ZSubgroupTree_fast    Examples:

HAP_SL2ZSubgroupTree_slow    Examples:

HAP_Sequence2Boundaries    Examples:

HAP_SimplicialPairToCWMap    Examples:

HAP_SimplicialProjectivePlane    Examples:

HAP_SimplicialTorus    Examples:

HAP_SimplifiedGaussCode    Examples:

HAP_StiefelWhitney    Examples:

HAP_SylowSubgroups    Examples:

HAP_Tensor    Examples:

HAP_TransversalCongruenceSubgroups    Examples:

HAP_TransversalCongruenceSubgroupsIdeal    Examples:

HAP_TransversalCongruenceSubgroupsIdeal_alt    Examples:

HAP_TransversalGamma0SubgroupsIdeal    Examples:

HAP_Triangulation    Examples:

HAP_TzPair    Examples:

HAP_WedgeSumOfSimplicialComplexes    Examples:

HAP_bockstein    Examples:

HAP_chain_bockstein    Examples:

HAP_coho_isoms    Examples:

HAP_nxnMatTo2nx2nMat    Examples:

HadamardGraph    Examples:

HapExample    Examples:

HapFile    Examples: 1 , 2 , 3 , 4 

HasTrivialPostnikovInvariant    Examples:

HeckeOperator    Examples:

HeckeOperatorWeight2    Examples:

HenonOrbit    Examples: 1 

HomToGModule_hom    Examples:

HomToInt_ChainComplex    Examples:

HomToInt_ChainMap    Examples:

HomToInt_CochainComplex    Examples:

HomToModPModule    Examples: 1 

HomogeneousPolynomials    Examples:

HomogeneousPolynomials_Bianchi    Examples:

HomologicalGroupDecomposition    Examples: 1 

HomologyOfPureCubicalComplex    Examples:

HomologyPbs    Examples:

HomologySimplicialFreeAbelianGroup    Examples:

HomomorphismAsMatrix    Examples:

HomotopyCatOneGroup    Examples:

HomotopyCrossedModule    Examples:

HomotopyEquivalentLargerSubArray    Examples:

HomotopyEquivalentLargerSubArray3D    Examples:

HomotopyEquivalentLargerSubMatrix    Examples:

HomotopyEquivalentLargerSubPermArray    Examples:

HomotopyEquivalentLargerSubPermArray3D    Examples:

HomotopyEquivalentLargerSubPermMatrix    Examples:

HomotopyEquivalentMaximalPureSubcomplex    Examples:

HomotopyEquivalentMinimalPureSubcomplex    Examples:

HomotopyEquivalentSmallerSubArray    Examples:

HomotopyEquivalentSmallerSubArray3D    Examples:

HomotopyEquivalentSmallerSubMatrix    Examples:

HomotopyEquivalentSmallerSubPermArray    Examples:

HomotopyEquivalentSmallerSubPermArray3D    Examples:

HomotopyEquivalentSmallerSubPermMatrix    Examples:

HomotopyLowerCentralSeries    Examples:

HomotopyLowerCentralSeriesOfCrossedModule    Examples:

HomotopyTruncation    Examples:

HopfSatohSurface    Examples: 1 , 2 

HybridSubdivision    Examples:

IdCatOneGroup    Examples: 1 

IdCrossedModule    Examples:

IdQuasiCatOneGroup    Examples:

IdQuasiCrossedModule    Examples:

IdentifyKnot    Examples: 1 

IdentityAmongRelators    Examples: 1 , 2 , 3 

ImageOfGOuterGroupHomomorphism    Examples: 1 , 2 

ImageOfMap    Examples:

InducedSteenrodHomomorphisms    Examples:

IntegerSimplicialComplex    Examples: 1 

IntegralCellularHomology    Examples:

IntegralCohomology    Examples:

IntegralCohomologyOfCochainComplex    Examples:

IntegralHomology    Examples: 1 

IntegralHomologyOfChainComplex    Examples:

IntersectionCWSubcomplex    Examples:

IsClosedManifold    Examples: 1 

IsContractibleCube_higherdims    Examples:

IsCrystSameOrbit    Examples:

IsCrystSufficientLattice    Examples:

IsHadamardMatrix    Examples:

IsIntList    Examples:

IsIsomorphismOfAbelianFpGroups    Examples: 1 

IsMetricMatrix    Examples:

IsPeriodicSpaceGroup    Examples: 1 

IsPureComplex    Examples:

IsPureRegularCWComplex    Examples:

IsRigid    Examples: 1 

IsRigidOnRight    Examples:

IsSphericalCoxeterGroup    Examples:

IsoclinismClasses    Examples: 1 , 2 

IsomorphismCatOneGroups    Examples: 1 

IsomorphismCrossedModules    Examples:

KernelOfGOuterGroupHomomorphism    Examples: 1 , 2 

KernelOfMap    Examples:

KernelWG    Examples:

KinkArc2Presentation    Examples:

KnotComplement    Examples: 1 , 2 , 3 

KnotComplementWithBoundary    Examples: 1 , 2 , 3 

LazyList    Examples:

LefschetzNumberOfChainMap    Examples:

Lfunction    Examples:

LiftColouredSurface    Examples:

LiftedRegularCWMap    Examples:

LinearHomomorphismsZZPersistenceMat    Examples:

LinkingForm    Examples: 1 

LinkingFormHomeomorphismInvariant    Examples: 1 

LinkingFormHomotopyInvariant    Examples: 1 

ListsOfCellsToRegularCWComplex    Examples:

LowDimensionalCupProduct    Examples: 1 

MakeHAPprimeDoc    Examples:

ManifoldType    Examples: 1 

Mapper    Examples: 1 

Mapper_alt    Examples:

MatrixSize    Examples:

MaximalSimplicesOfSimplicialComplex    Examples: 1 

MaximalSphericalCoxeterSubgroupsFromAbove    Examples:

MinimizeRingRelations    Examples:

Mod2SteenrodAlgebra    Examples: 1 

ModPCohomologyRing_alt    Examples:

ModPCohomologyRing_part_1    Examples:

ModPCohomologyRing_part_2    Examples:

ModPRingGeneratorsAlt    Examples:

ModPSteenrodAlgebra    Examples: 1 , 2 

ModularCohomology    Examples:

ModularEquivariantChainMap    Examples:

ModularHomology    Examples:

Nil3TensorSquare    Examples:

NonFreeResolutionFiniteSubgroup    Examples:

NonManifoldVertices    Examples:

NonRegularCWBoundary    Examples:

NonabelianSymmetricKernel_alt    Examples: 1 

NonabelianSymmetricSquare_inf    Examples:

NonabelianTensorProduct_Inf    Examples:

NonabelianTensorProduct_alt    Examples:

NonabelianTensorSquareAsCatOneGroup    Examples:

NonabelianTensorSquareAsCrossedModule    Examples:

NonabelianTensorSquare_inf    Examples:

NoncrossingPartitionsLatticeDisplay    Examples: 1 

NullspaceSparseMatDestructive    Examples:

NumberConnectedQuandles    Examples:

NumberGeneratorsOfGroupHomology    Examples:

NumberOfCrossingsInArc2Presentation    Examples:

NumberOfHomomorphisms_connected    Examples:

NumberOfHomomorphisms_groups    Examples:

NumberOfPrimeKnots    Examples: 1 , 2 

NumberSmallCatOneGroups    Examples:

NumberSmallCrossedModules    Examples:

NumberSmallQuasiCatOneGroups    Examples:

NumberSmallQuasiCrossedModules    Examples:

OppositeGroup    Examples:

OrthogonalizeBasisByAverageInnerProduct    Examples:

PCentre    Examples:

PSubgroupGChainComplex    Examples:

PSubgroupSimplicialComplex    Examples:

PUpperCentralSeries    Examples:

PartialIsoclinismClasses    Examples: 1 

PartsOfQuadraticInteger    Examples:

PathComponentOfPureComplex    Examples: 1 

PathComponentsCWSubcomplex    Examples:

PathComponentsOfSimplicialComplex_alt    Examples:

PathObjectForChainComplex    Examples: 1 

PermutahedralComplexToRegularCWComplex    Examples: 1 

PermutahedralToCubicalArray    Examples:

PersistentBettiNumbersViaContractions    Examples:

PersistentHomologyOfCrossedModule    Examples:

PersistentHomologyOfFilteredPureCubicalComplex_alt    Examples:

PersistentHomologyOfFilteredSparseChainComplex    Examples: 1 , 2 

PersistentHomologyOfPureCubicalComplex_Alt    Examples:

PersistentHomologyOfQuotientGroupSeries_Int    Examples:

PiZeroOfRegularCWComplex    Examples:

PoincareBipyramidCWComplex    Examples: 1 

PoincareCubeCWComplex    Examples: 1 

PoincareCubeCWComplexNS    Examples: 1 

PoincareDodecahedronCWComplex    Examples: 1 , 2 

PoincareOctahedronCWComplex    Examples: 1 

PoincarePrismCWComplex    Examples: 1 

PoincareSeriesApproximation    Examples:

PoincareSeries_alt    Examples:

PolymakeFaceLattice    Examples:

PolytopalRepresentationComplex    Examples:

PrankAlt    Examples:

PresentationOfResolution_alt    Examples:

PrimePartDerivedFunctorHomomorphism    Examples:

PrimePartDerivedFunctorViaSubgroupChain    Examples:

PrimePartDerivedTwistedFunctor    Examples:

PrintAlgebraWordAsPolynomial    Examples:

PrintTorsionSubcomplex    Examples:

PureComplex    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 

PureCubicalComplexToCubicalComplex    Examples: 1 , 2 

PureCubicalLink    Examples: 1 , 2 

PushoutOfFpGroups    Examples:

QuadraticCharacter    Examples:

QuadraticNumberField    Examples: 1 

QuandleIsomorphismRepresentatives    Examples:

QuotientByTorsionSubcomplex    Examples:

QuotientChainMap    Examples:

QuotientGroup    Examples:

QuotientQuasiIsomorph    Examples:

RadicalSeriesOfResolution    Examples:

RandomArc2Presentation    Examples:

RandomCellOfPureComplex    Examples:

ReadLinkImageAsGaussCode    Examples: 1 

ReadMatrixAsPureCubicalComplex    Examples:

ReduceGenerators    Examples:

ReduceGenerators_alt    Examples:

ReflectedCubicalKnot    Examples: 1 , 2 , 3 , 4 

RegularCWAssociahedron    Examples:

RegularCWComplexComplement    Examples: 1 

RegularCWComplexWithRemovedCell    Examples: 1 

RegularCWComplex_AttachCellDestructive    Examples: 1 

RegularCWCube    Examples:

RegularCWMapToCWSubcomplex    Examples:

RegularCWOrbitPolytope    Examples:

RegularCWPermutahedron    Examples:

RegularCWPolygon    Examples:

RegularCWSimplex    Examples:

RelativeCentralQuotientSpaceGroup    Examples:

RelativeGroupHomology    Examples:

RelativeRightTransversal    Examples:

RemoveStar    Examples:

ResolutionAbelianGroup_alt    Examples:

ResolutionAbelianPcpGroup    Examples:

ResolutionAffineCrystGroup    Examples:

ResolutionBoundaryOfWordOnRight    Examples:

ResolutionDirectProductLazy    Examples:

ResolutionFiniteCyclicGroup    Examples:

ResolutionGL2QuadraticIntegers    Examples:

ResolutionGL3QuadraticIntegers    Examples:

ResolutionGenericGroup    Examples:

ResolutionInfiniteCyclicGroup    Examples:

ResolutionPGL2QuadraticIntegers    Examples:

ResolutionPGL3QuadraticIntegers    Examples:

ResolutionPSL2QuadraticIntegers    Examples: 1 

ResolutionPrimePowerGroupSparse    Examples:

ResolutionSL2QuadraticIntegers    Examples: 1 

ResolutionSL2ZConjugated    Examples:

ResolutionSL2Z_alt    Examples:

ResolutionSpaceGroup    Examples: 1 

ResolutionToEquivariantCWComplex    Examples:

ResolutionToResolutionOfFpGroup    Examples: 1 

SL2QuadraticIntegers    Examples: 1 

SL2ZResolution    Examples:

SL2ZResolution_alt    Examples:

SL2ZTree    Examples:

SL2ZmElementsDecomposition    Examples:

SequentialRegularCWComplexComplement    Examples:

SignatureOfSymmetricMatrix    Examples: 1 

SignedPermutationGroup    Examples: 1 

SimplicesToSimplicialComplex    Examples: 1 , 2 , 3 , 4 

SimplicialComplexToRegularCWComplex_alt    Examples:

SimplicialK3Surface    Examples: 1 

SimplicialNerveOfFilteredGraph    Examples: 1 , 2 

SimplicialNerveOfTwoComplex    Examples:

SimplifiedQuandlePresentation    Examples:

SimplifiedRegularCWComplex    Examples: 1 

SimplifiedSparseChainComplex    Examples:

SmallCatOneGroup    Examples: 1 

SmallCrossedModule    Examples:

SmallQuasiCatOneGroup    Examples:

SmallQuasiCrossedModule    Examples:

SmoothedFpGroup    Examples:

SparseChainComplexOfCubicalComplex    Examples:

SparseChainComplexOfCubicalPair    Examples:

SparseChainComplexOfFilteredRegularCWComplex    Examples:

SparseChainComplexOfRegularCWComplexWithVectorField    Examples:

SparseChainComplexOfSimplicialComplex    Examples:

SparseChainComplexToChainComplex    Examples:

SparseChainMapOfCubicalPairs    Examples:

SparseFilteredChainComplexOfFilteredCubicalComplex    Examples:

SparseFilteredChainComplexOfFilteredSimplicialComplex    Examples: 1 , 2 

SparseMattoMat    Examples: 1 

SparseRowReduce    Examples:

SphericalKnotComplement    Examples: 1 

Spin    Examples:

SpunAboutHyperplane    Examples:

SpunKnotComplement    Examples: 1 

SpunLinkComplement    Examples:

StrongGeneratorsOfDerivedSubgroup    Examples:

StrongGeneratorsOfDerivedSubgroup_alt    Examples:

StructuralCopyOfFilteredRegularCWComplex    Examples:

SubQuasiIsomorph    Examples:

SubdivideCell    Examples:

Suspension_alt    Examples:

SylowSubgroupOfCatOneGroup    Examples:

SymmetricCentre    Examples:

SymmetricCommutativityGroup    Examples:

TensorNonFreeResolutionWithRationals    Examples:

TensorWithBurnsideRing    Examples: 1 , 2 

TensorWithComplexRepresentationRing    Examples: 1 , 2 

TensorWithComplexRepresentationRingOnRight    Examples:

TensorWithIntegersModPSparse    Examples:

TensorWithIntegersOverSubgroup    Examples: 1 , 2 , 3 , 4 

TensorWithIntegersSparse    Examples:

TensorWithModPModule    Examples: 1 

TestHapBook    Examples:

TestHapQuick    Examples:

ThickenedHEPureCubicalComplex    Examples:

ThickenedPureComplex    Examples: 1 

ThickenedPureCubicalComplex_dim2    Examples:

ThirdHomotopyGroupOfSuspensionB_alt    Examples: 1 

ThreeManifoldViaDehnSurgery    Examples: 1 

ThreeManifoldWithBoundary    Examples: 1 

TransferChainMap    Examples: 1 

TransferCochainMap    Examples: 1 

TranslationSubGroup    Examples:

TreeOfResolutionsToSL2Zcomplex    Examples:

TruncatedRegularCWComplex    Examples:

Tube    Examples:

TupleOrbitReps    Examples:

TupleOrbitReps_perm    Examples:

TwistedResolution    Examples:

UnboundedArrayAssign    Examples:

UnitBall    Examples:

UnitCubicalBall    Examples:

UnitPermutahedralBall    Examples:

UniversalBarCodeEval    Examples:

UniversalCover    Examples: 1 , 2 , 3 , 4 

VectorToCrystMatrix    Examples:

VectorsToOneSkeleton    Examples: 1 

VerticesOfRegularCWCell    Examples:

View3dPureComplex    Examples:

ViewArc2Presentation    Examples:

ViewPureComplex    Examples:

VirtuallySimplicialSubdivision    Examples:

WeakCommutativityGroup    Examples:

WirtingerGroup    Examples: 1 

WirtingerGroup_gc    Examples:

WordModP    Examples:

ZigZagContractedFilteredPureCubicalComplex    Examples:

ZigZagContractedPureComplex    Examples: 1 

Sq    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 

Category_Of_Groups    Examples: 1 

PreImagesElmNC    Examples:

PreImagesNC    Examples:

PreImagesSetNC    Examples:

AsFpGroup    Examples:

BarycentricSubdivision    Examples: 1 , 2 

Bockstein    Examples: 1 , 2 , 3 

CategoryArrow    Examples:

CategoryObject    Examples:

ClosedSurface    Examples: 1 

CoboundaryMatrix    Examples:

CoefficientsOfPoincareSeries    Examples:

CohomologyClass    Examples: 1 , 2 

CohomologyRing    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 

ComplexProjectiveSpace    Examples: 1 

CompositionRingHomomorphism    Examples:

ConnectedComponentsQuandle    Examples:

ConnectedSum    Examples: 1 , 2 

DegreeOfRepresentative    Examples:

Dimensions    Examples:

ExcisedPair    Examples:

ExpandedComplex    Examples: 1 

FilteredRegularCWComplex    Examples: 1 

FundamentalGroupWithPathReps    Examples: 1 , 2 

GDerivedSubgroup    Examples:

GModuleAsGOuterGroup    Examples: 1 , 2 , 3 , 4 

GOuterGroupHomomorphism    Examples: 1 , 2 

GOuterGroupHomomorphism    Examples: 1 , 2 

GradedAlgebraPresentation    Examples:

GradedAlgebraPresentationNC    Examples:

HAPDerivationNC    Examples:

HAPRingHomomorphismByIndeterminateMap    Examples:

HAPRingReductionHomomorphism    Examples:

HAPRingReductionHomomorphism    Examples:

HAPRingToSubringHomomorphism    Examples:

HAPSubringToRingHomomorphism    Examples:

HAPSubringToRingHomomorphism    Examples:

HAPZeroRingHomomorphism    Examples:

HAP_EquivalenceClasses    Examples:

HomomorphismsImages    Examples:

ImageOfDerivation    Examples:

ImageOfRingHomomorphism    Examples:

IsAssociatedGradedRing    Examples:

KernelOfDerivation    Examples:

LowerGCentralSeries    Examples:

PathComponents    Examples: 1 

PersistentBettiNumbersAlt    Examples: 1 

PersistentHomology    Examples: 1 , 2 

PersistentHomology    Examples: 1 , 2 

PersistentHomology    Examples: 1 , 2 

PoincareSeriesAutoMem    Examples:

PoincareSeriesAutoMem    Examples:

PoincareSeriesAutoMemStop    Examples:

PolynomialToRModuleRep    Examples:

PreimageOfRingHomomorphism    Examples:

PureComplexMeet    Examples:

PureComplexRandomCell    Examples: 1 

PureComplexSubcomplex    Examples:

Pushout    Examples:

QuadraticIdeal    Examples: 1 

ReduceIdeal    Examples:

ReducedPolynomialRingPresentation    Examples:

ReducedPolynomialRingPresentationMap    Examples:

RefinedColouring    Examples:

Resolution    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 

RightTransversal_alt    Examples:

RingOfIntegers    Examples: 1 

SingularPolynomialNormalForm    Examples:

SingularSetNormalFormIdeal    Examples:

SingularSetNormalFormIdealNC    Examples:

SparseChainComplexOfPair    Examples:

Sphere    Examples: 1 

Sq    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 

Standard2Cocycle    Examples:

Standard2Cocycle    Examples:

StandardNCocycle    Examples:

StandardNCocycle    Examples:

StarGraph    Examples:

SubspaceBasisRepsByDegree    Examples:

SubspaceDimensionDegree    Examples:

Suspension    Examples: 1 , 2 , 3 , 4 , 5 

TrivialGModuleAsGOuterGroup    Examples: 1 , 2 , 3 , 4 

VertexLink    Examples:

VertexStar    Examples:

WedgeSum    Examples: 1 

TensorProductOp    Examples:

Arity    Examples:

AssociatedNumberField    Examples:

AssociatedRing    Examples:

Base    Examples: 1 

BaseElement    Examples:

BaseRing    Examples:

Cocycle    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 

CoefficientModule    Examples:

CohomologicalPeriod    Examples: 1 

CoxeterMatrix    Examples: 1 

DerivationImages    Examples:

DerivationRelations    Examples:

DerivationRing    Examples:

Fibre    Examples:

FibreElement    Examples:

Generators    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 

GeneratorsOfPresentationIdeal    Examples:

GradedAlgebraPresentationFamily    Examples:

HAPDerivationFamily    Examples:

HAPPRIME_HilbertSeries    Examples:

HAPRingHomomorphismFamily    Examples:

HAP_MultiplicativeGenerators    Examples:

IdentityMap    Examples:

ImageGenerators    Examples:

ImagePolynomialRing    Examples:

ImageRelations    Examples:

InCcGroup    Examples:

IndeterminateAndExponentOfUnivariateMonomial    Examples:

IndeterminateDegrees    Examples:

IndeterminatesOfGradedAlgebraPresentation    Examples:

IndeterminatesOfPolynomial    Examples:

IndexInSL2O    Examples: 1 

InnerAutomorphismGroupQuandle    Examples:

InnerAutomorphismGroupQuandleAsPerm    Examples:

InverseRingHomomorphism    Examples:

IsConnected    Examples: 1 , 2 , 3 

IsHomogeneousQuandle    Examples:

IsLatinQuandle    Examples: 1 

MaximumDegreeForPresentation    Examples:

ModPRingBasisAsPolynomials    Examples:

ModPRingGeneratorDegrees    Examples:

ModPRingNiceBasis    Examples:

ModPRingNiceBasisAsPolynomials    Examples:

Module    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 

NormOfIdeal    Examples:

OuterAction    Examples:

OuterGroup    Examples: 1 , 2 , 3 , 4 

PresentationIdeal    Examples:

PresentationOfGradedStructureConstantAlgebra    Examples: 1 

Pullbacks    Examples:

Pushouts    Examples:

RightMultiplicationGroupOfQuandle    Examples: 1 , 2 , 3 

RightMultiplicationGroupOfQuandleAsPerm    Examples: 1 

SingularGroebnerBasis    Examples:

SingularReducedGroebnerBasis    Examples:

SourceGenerators    Examples:

SourcePolynomialRing    Examples:

SourceRelations    Examples:

StarGraphAttr    Examples:

TermsOfPolynomial    Examples:

UnivariateMonomialsOfMonomial    Examples:

CoefficientsRing    Examples:

ElementsFamily    Examples:

IndexInSL2Z    Examples:

Name    Examples: 1 , 2 , 3 , 4 , 5 , 6 

Order    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 

IsAbelianCategory    Examples:

IsAdditiveCategory    Examples:

IsCategoryName    Examples:

IsCcGroup    Examples:

IsCrystTranslationSubGroup    Examples:

IsGOuterGroup    Examples:

IsGOuterGroupHomomorphism    Examples:

IsGammaSubgroupInSL3Z    Examples:

IsHAPRationalMatrixGroup    Examples:

IsHAPRationalSpecialLinearGroup    Examples:

IsIdealOfQuadraticIntegers    Examples:

IsPeriodic    Examples: 1 , 2 

IsPseudoListWithFunction    Examples:

IsQuadraticNumberField    Examples:

IsRingOfQuadraticIntegers    Examples:

IsStandard2Cocycle    Examples:

IsStandardNCocycle    Examples:

IsCcElement    Examples:

IsGradedAlgebraPresentation    Examples:

IsHAPDerivation    Examples:

IsHAPRingHomomorphism    Examples:

IsHAPRingModIdealObj    Examples:

IsHapCatOneGroup    Examples:

IsHapCatOneGroupMorphism    Examples:

IsHapChainComplex    Examples:

IsHapChainMap    Examples:

IsHapCochainComplex    Examples:

IsHapCochainMap    Examples:

IsHapCommutativeDiagram    Examples:

IsHapConjQuandElt    Examples:

IsHapCrossedModule    Examples:

IsHapCrossedModuleMorphism    Examples:

IsHapCubicalComplex    Examples:

IsHapEquivariantCWComplex    Examples:

IsHapEquivariantChainComplex    Examples:

IsHapEquivariantChainMap    Examples:

IsHapEquivariantNonFreeChainComplex    Examples:

IsHapEquivariantSpectralSequencePage    Examples:

IsHapFilteredChainComplex    Examples:

IsHapFilteredCubicalComplex    Examples:

IsHapFilteredGraph    Examples:

IsHapFilteredPureCubicalComplex    Examples:

IsHapFilteredRegularCWComplex    Examples:

IsHapFilteredSimplicialComplex    Examples:

IsHapFilteredSparseChainComplex    Examples:

IsHapGCocomplex    Examples:

IsHapGComplex    Examples:

IsHapGComplexMap    Examples:

IsHapGraph    Examples:

IsHapOppositeElement    Examples:

IsHapPureCubicalComplex    Examples:

IsHapPureCubicalLink    Examples:

IsHapPurePermutahedralComplex    Examples:

IsHapQuandlePresentation    Examples:

IsHapQuotientElement    Examples:

IsHapRegularCWComplex    Examples:

IsHapRegularCWMap    Examples:

IsHapResolution    Examples:

IsHapSimplicialComplex    Examples:

IsHapSimplicialFreeAbelianGroup    Examples:

IsHapSimplicialGroup    Examples:

IsHapSimplicialGroupMorphism    Examples:

IsHapSimplicialMap    Examples:

IsHapSparseChainComplex    Examples:

IsHapSparseChainMap    Examples:

IsHapSparseMat    Examples:

IsHapTorsionSubcomplex    Examples:

IsPseudoList    Examples:

IsCcElementRep    Examples:

IsGradedAlgebraPresentationRep    Examples:

IsHAPDerivationRep    Examples:

IsHAPIdealRep    Examples:

IsHAPRingHomomorphismIndeterminateMapRep    Examples:

IsHAPRingReductionHomomorphismRep    Examples:

IsHAPRingToSubringHomomorphismRep    Examples:

IsHAPSubringToRingHomomorphismRep    Examples:

IsHAPZeroRingHomomorphismRep    Examples:

IsHapCatOneGroupMorphismRep    Examples:

IsHapCatOneGroupRep    Examples:

IsHapChainComplexRep    Examples:

IsHapChainMapRep    Examples:

IsHapCochainComplexRep    Examples:

IsHapCochainMapRep    Examples:

IsHapCommutativeDiagramRep    Examples:

IsHapConjQuandEltRep    Examples:

IsHapCrossedModuleMorphismRep    Examples:

IsHapCrossedModuleRep    Examples:

IsHapCubicalComplexRep    Examples:

IsHapEquivariantCWComplexRep    Examples:

IsHapEquivariantChainComplexRep    Examples:

IsHapEquivariantChainMapRep    Examples:

IsHapEquivariantNonFreeChainComplexRep    Examples:

IsHapEquivariantSpectralSequencePageRep    Examples:

IsHapFilteredChainComplexRep    Examples:

IsHapFilteredCubicalComplexRep    Examples:

IsHapFilteredGraphRep    Examples:

IsHapFilteredPureCubicalComplexRep    Examples:

IsHapFilteredRegularCWComplexRep    Examples:

IsHapFilteredSimplicialComplexRep    Examples:

IsHapFilteredSparseChainComplexRep    Examples:

IsHapGCocomplexRep    Examples:

IsHapGComplexMapRep    Examples:

IsHapGComplexRep    Examples:

IsHapGraphRep    Examples:

IsHapOppositeElementRep    Examples:

IsHapPureCubicalComplexRep    Examples:

IsHapPureCubicalLinkRep    Examples:

IsHapPurePermutahedralComplexRep    Examples:

IsHapQuandlePresentationRep    Examples:

IsHapQuotientElementRep    Examples:

IsHapRegularCWComplexRep    Examples:

IsHapRegularCWMapRep    Examples:

IsHapResolutionRep    Examples:

IsHapSimplicialComplexRep    Examples:

IsHapSimplicialFreeAbelianGroupRep    Examples:

IsHapSimplicialGroupMorphismRep    Examples:

IsHapSimplicialGroupRep    Examples:

IsHapSimplicialMapRep    Examples:

IsHapSparseChainComplexRep    Examples:

IsHapSparseChainMapRep    Examples:

IsHapSparseMatRep    Examples:

IsHapTorsionSubcomplexRep    Examples:

IsPseudoListRep    Examples:

IdealOfQuadraticIntegers    Examples:

QuadraticNF    Examples:

RingOfQuadraticIntegers    Examples:

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

+    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 

+    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 

+    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 

    Examples:

    Examples:

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

AdditiveInverseMutable    Examples:

AsFpGroup    Examples:

AsList    Examples:

AsSSortedList    Examples:

BarycentricSubdivision    Examples: 1 , 2 

BaseRing    Examples:

Bockstein    Examples: 1 , 2 , 3 

CategoryArrow    Examples:

CategoryObject    Examples:

CoboundaryMatrix    Examples:

CoefficientsOfPoincareSeries    Examples:

CoefficientsRing    Examples:

CohomologicalPeriod    Examples: 1 

CohomologyClass    Examples: 1 , 2 

CompositionRingHomomorphism    Examples:

CompositionRingHomomorphism    Examples:

CompositionRingHomomorphism    Examples:

CompositionRingHomomorphism    Examples:

CompositionRingHomomorphism    Examples:

ConnectedComponentsQuandle    Examples:

ConnectedSum    Examples: 1 , 2 

CoxeterMatrix    Examples: 1 

DefaultFieldOfMatrixGroup    Examples:

DegreeOfRepresentative    Examples:

DerivationImages    Examples:

DerivationRelations    Examples:

DerivationRing    Examples:

Dimensions    Examples:

Enumerator    Examples:

ExcisedPair    Examples:

FilteredRegularCWComplex    Examples: 1 

FundamentalGroupWithPathReps    Examples: 1 , 2 

GModuleAsGOuterGroup    Examples: 1 , 2 , 3 , 4 

GOuterGroupHomomorphism    Examples: 1 , 2 

GOuterGroupHomomorphism    Examples: 1 , 2 

Generators    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 

GeneratorsOfMagmaWithInverses    Examples:

GeneratorsOfMagmaWithInverses    Examples:

GeneratorsOfPresentationIdeal    Examples:

GradedAlgebraPresentation    Examples:

GradedAlgebraPresentationNC    Examples:

HAPDerivationNC    Examples:

HAPPRIME_HilbertSeries    Examples:

HAPRingHomomorphismByIndeterminateMap    Examples:

HAPRingReductionHomomorphism    Examples:

HAPRingReductionHomomorphism    Examples:

HAPRingToSubringHomomorphism    Examples:

HAPSubringToRingHomomorphism    Examples:

HAPSubringToRingHomomorphism    Examples:

HAPZeroRingHomomorphism    Examples:

HAP_MultiplicativeGenerators    Examples:

HomomorphismsImages    Examples:

IdGroup    Examples: 1 , 2 , 3 , 4 , 5 , 6 

IdentityMap    Examples:

ImageOfDerivation    Examples:

ImageOfRingHomomorphism    Examples:

ImageOfRingHomomorphism    Examples:

ImageOfRingHomomorphism    Examples:

ImageOfRingHomomorphism    Examples:

ImageOfRingHomomorphism    Examples:

IndeterminateAndExponentOfUnivariateMonomial    Examples:

IndeterminateDegrees    Examples:

IndeterminatesOfGradedAlgebraPresentation    Examples:

IndeterminatesOfPolynomial    Examples:

IndexInSL2O    Examples: 1 

IndexInSL2Z    Examples:

InnerAutomorphismGroupQuandle    Examples:

InnerAutomorphismGroupQuandleAsPerm    Examples:

Int    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

InverseMutable    Examples:

InverseMutable    Examples:

InverseMutable    Examples:

InverseRingHomomorphism    Examples:

InverseRingHomomorphism    Examples:

InverseRingHomomorphism    Examples:

InverseRingHomomorphism    Examples:

InverseRingHomomorphism    Examples:

InverseSameMutability    Examples:

IsAssociatedGradedRing    Examples:

IsConnected    Examples: 1 , 2 , 3 

IsHomogeneousQuandle    Examples:

IsLatinQuandle    Examples: 1 

IsMonomial    Examples:

IsOne    Examples:

IsPeriodic    Examples: 1 , 2 

Kernel    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 

KernelOfDerivation    Examples:

MaximumDegreeForPresentation    Examples:

ModPRingBasisAsPolynomials    Examples:

ModPRingGeneratorDegrees    Examples:

ModPRingNiceBasis    Examples:

ModPRingNiceBasisAsPolynomials    Examples:

OneImmutable    Examples:

OneMutable    Examples:

PathComponents    Examples: 1 

PersistentBettiNumbersAlt    Examples: 1 

PreimageOfRingHomomorphism    Examples:

PresentationIdeal    Examples:

PresentationOfGradedStructureConstantAlgebra    Examples: 1 

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

PrintObj    Examples:

Projection    Examples:

PureComplexMeet    Examples:

PureComplexRandomCell    Examples: 1 

PureComplexSubcomplex    Examples:

Pushout    Examples:

QuadraticIdeal    Examples: 1 

Random    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 

ReduceIdeal    Examples:

ReducedPolynomialRingPresentation    Examples:

ReducedPolynomialRingPresentationMap    Examples:

RefinedColouring    Examples:

Resolution    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 

RightMultiplicationGroupOfQuandle    Examples: 1 , 2 , 3 

RightMultiplicationGroupOfQuandleAsPerm    Examples: 1 

RightTransversal    Examples:

RingOfIntegers    Examples: 1 

SingularGroebnerBasis    Examples:

SingularPolynomialNormalForm    Examples:

SingularReducedGroebnerBasis    Examples:

SingularSetNormalFormIdeal    Examples:

SingularSetNormalFormIdealNC    Examples:

SparseChainComplexOfPair    Examples:

Sq    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 

Standard2Cocycle    Examples:

Standard2Cocycle    Examples:

StandardNCocycle    Examples:

StandardNCocycle    Examples:

StarGraph    Examples:

StarGraphAttr    Examples:

SubspaceBasisRepsByDegree    Examples:

SubspaceDimensionDegree    Examples:

Suspension    Examples: 1 , 2 , 3 , 4 , 5 

TensorProductOp    Examples:

TermsOfPolynomial    Examples:

TrivialGModuleAsGOuterGroup    Examples: 1 , 2 , 3 , 4 

Units    Examples:

Units    Examples:

UnivariateMonomialsOfMonomial    Examples:

VertexLink    Examples:

VertexStar    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

ViewObj    Examples:

WedgeSum    Examples: 1 

ZeroMutable    Examples:

^    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 

^    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 

^    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

mod    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

*    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 

+    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 

+    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 

+    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 

-    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

-    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

/    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

    Examples:

    Examples:

    Examples:

    Examples:

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

=    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

AbelianInvariants    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 

AbelianInvariants    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 

BarycentricSubdivision    Examples: 1 , 2 

Bockstein    Examples: 1 , 2 , 3 

CanonicalRightCosetElement    Examples:

ClosedSurface    Examples: 1 

CohomologyRing    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 

CohomologyRing    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 

ComplexProjectiveSpace    Examples: 1 

ConnectedSum    Examples: 1 , 2 

ConnectedSum    Examples: 1 , 2 

ConnectedSum    Examples: 1 , 2 

Dimensions    Examples:

DirectProductOp    Examples:

DirectProductOp    Examples:

DirectProductOp    Examples:

DirectProductOp    Examples:

Discriminant    Examples:

Discriminant    Examples:

Embedding    Examples:

ExpandedComplex    Examples: 1 

ExpandedComplex    Examples: 1 

ExpandedComplex    Examples: 1 

ExpandedComplex    Examples: 1 

GDerivedSubgroup    Examples:

Generators    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 

HAPRingReductionHomomorphism    Examples:

HAPRingReductionHomomorphism    Examples:

HAP_EquivalenceClasses    Examples:

ImageOfRingHomomorphism    Examples:

ImageOfRingHomomorphism    Examples:

IndexNC    Examples:

IndexNC    Examples:

InverseMutable    Examples:

InverseMutable    Examples:

InverseMutable    Examples:

InverseSameMutability    Examples:

IsEmpty    Examples:

IsEmpty    Examples:

IsPrime    Examples: 1 , 2 

IsomorphismFpGroup    Examples: 1 , 2 

Iterator    Examples:

KernelOfDerivation    Examples:

ListOp    Examples:

ListOp    Examples:

LowerGCentralSeries    Examples:

NaturalHomomorphism    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 

Norm    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 

Norm    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 

OneImmutable    Examples:

OneImmutable    Examples:

Order    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 

Order    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 

PersistentBettiNumbersAlt    Examples: 1 

PersistentBettiNumbersAlt    Examples: 1 

PersistentBettiNumbersAlt    Examples: 1 

PersistentBettiNumbersAlt    Examples: 1 

PersistentBettiNumbersAlt    Examples: 1 

PersistentHomology    Examples: 1 , 2 

PersistentHomology    Examples: 1 , 2 

Position    Examples: 1 , 2 

Position    Examples: 1 , 2 

Position    Examples: 1 , 2 

PositionCanonical    Examples:

PreimageOfRingHomomorphism    Examples:

Projection    Examples:

PureComplexMeet    Examples:

PureComplexRandomCell    Examples: 1 

PureComplexSubcomplex    Examples:

QuadraticIdeal    Examples: 1 

Range    Examples: 1 , 2 

RankMatrixDestructive    Examples:

ReduceIdeal    Examples:

ReduceIdeal    Examples:

ReducedPolynomialRingPresentation    Examples:

ReducedPolynomialRingPresentation    Examples:

ReducedPolynomialRingPresentationMap    Examples:

ReducedPolynomialRingPresentationMap    Examples:

ReducedPolynomialRingPresentationMap    Examples:

RefinedColouring    Examples:

RightTransversal    Examples:

RightTransversal    Examples:

RightTransversal    Examples:

RightTransversal    Examples:

RightTransversal    Examples:

RightTransversal_alt    Examples:

SingularPolynomialNormalForm    Examples:

SparseChainComplexOfPair    Examples:

Sphere    Examples: 1 

SubspaceBasisRepsByDegree    Examples:

SubspaceDimensionDegree    Examples:

Suspension    Examples: 1 , 2 , 3 , 4 , 5 

TensorProductOp    Examples:

TensorProductOp    Examples:

TensorProductOp    Examples:

Trace    Examples:

Units    Examples:

WedgeSum    Examples: 1 

WedgeSum    Examples: 1 

WedgeSum    Examples: 1 

[]    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 

[]    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 

[]    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 

^    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 

^    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 

^    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 

^    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

in    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 

mod    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 

PathComponentOfSimplicialComplex    Examples:

ResolutionSL2ZInvertedInteger    Examples:

ViewGraph    Examples:

InfoHAPprime    Examples:

ASY_PATH    Examples:

AutomorphismGroupAsCrossedModule    Examples:

BROWSER_PATH    Examples:

CATONEGROUP_DATA_PERM    Examples:

CATONEGROUP_DATA_SIZE    Examples:

Cedric_PlanarDiagram    Examples:

ChildKill    Examples:

DISPLAY_PATH    Examples:

DOT_PATH    Examples:

FilteredSimplicialComplexToFilteredCWComplex    Examples:

GradedAlgebraPresentationType    Examples:

HAPTEMPORARYFUNCTION    Examples:

HAP_Knots    Examples:

HAP_ROOT    Examples:

HapCatOneGroup    Examples:

HapCatOneGroupFamily    Examples:

HapCatOneGroupMorphism    Examples:

HapCatOneGroupMorphismFamily    Examples:

HapChainComplex    Examples:

HapChainComplexFamily    Examples:

HapChainMap    Examples:

HapChainMapFamily    Examples:

HapCochainComplex    Examples:

HapCochainComplexFamily    Examples:

HapCochainMap    Examples:

HapCochainMapFamily    Examples:

HapCommutativeDiagram    Examples:

HapCommutativeDiagramFamily    Examples:

HapCrossedModule    Examples:

HapCrossedModuleFamily    Examples:

HapCrossedModuleMorphism    Examples:

HapCrossedModuleMorphismFamily    Examples:

HapCubicalComplex    Examples:

HapCubicalComplexFamily    Examples:

HapEquivariantCWComplex    Examples:

HapEquivariantCWComplexFamily    Examples:

HapEquivariantChainMap    Examples:

HapEquivariantChainMapFamily    Examples:

HapFPGModule    Examples:

HapFPGModuleHomomorphism    Examples:

HapFilteredChainComplex    Examples:

HapFilteredChainComplexFamily    Examples:

HapFilteredCubicalComplex    Examples:

HapFilteredCubicalComplexFamily    Examples:

HapFilteredGraph    Examples:

HapFilteredGraphFamily    Examples:

HapFilteredPureCubicalComplex    Examples:

HapFilteredPureCubicalComplexFamily    Examples:

HapFilteredRegularCWComplex    Examples:

HapFilteredRegularCWComplexFamily    Examples:

HapFilteredSimplicialComplex    Examples:

HapFilteredSimplicialComplexFamily    Examples:

HapFilteredSparseChainComplex    Examples:

HapFilteredSparseChainComplexFamily    Examples:

HapGChainComplex    Examples:

HapGCocomplex    Examples:

HapGCocomplexFamily    Examples:

HapGComplex    Examples:

HapGComplexFamily    Examples:

HapGlobalDeclarationsAreAlreadyLoaded    Examples:

HapGraph    Examples:

HapGraphFamily    Examples:

HapNonFreeResolution    Examples:

HapOppositeElement    Examples:

HapOppositeElementFamily    Examples:

HapPureCubicalComplex    Examples:

HapPureCubicalComplexFamily    Examples:

HapPureCubicalLink    Examples:

HapPureCubicalLinkFamily    Examples:

HapPurePermutahedralComplex    Examples:

HapPurePermutahedralComplexFamily    Examples:

HapQuotientElement    Examples:

HapQuotientElementFamily    Examples:

HapRegularCWComplex    Examples:

HapRegularCWComplexFamily    Examples:

HapRegularCWMap    Examples:

HapRegularCWMapFamily    Examples:

HapResolution    Examples:

HapResolutionFamily    Examples:

HapSimplicialComplex    Examples:

HapSimplicialComplexFamily    Examples:

HapSimplicialGroup    Examples:

HapSimplicialGroupFamily    Examples:

HapSimplicialGroupMorphism    Examples:

HapSimplicialGroupMorphismFamily    Examples:

HapSimplicialMap    Examples:

HapSimplicialMapFamily    Examples:

HapSparseChainComplex    Examples:

HapSparseChainComplexFamily    Examples:

HapSparseChainMap    Examples:

HapSparseChainMapFamily    Examples:

HapSparseMat    Examples:

HapSparseMatFamily    Examples:

HomomorphismOfDirectProduct    Examples:

IDQUASICATONEGROUP_DATA    Examples:

IsHapChain    Examples:

IsHapCochain    Examples:

IsHapComplex    Examples:

IsHapFPGModule    Examples:

IsHapFPGModuleHomomorphism    Examples:

IsHapGChainComplex    Examples:

IsHapMap    Examples:

IsHapNonFreeResolution    Examples:

NEATO_PATH    Examples:

NerveOfCover    Examples:

POLYMAKE_PATH    Examples:

PseudoList    Examples:

PseudoListFamily    Examples:

QUASICATONEGROUP_DATA_NOT    Examples:

QUASICATONEGROUP_DATA_SIZE    Examples:

ReadBioData    Examples:

SMALLQUASICATONEGROUP_DATA    Examples:

CATONEGROUP_DATA    Examples:

COMPILED    Examples:

Cedric_XYXYConnQuan    Examples:

Cedric_XYXYQuandles    Examples:

CommutingProbability    Examples:

GroupIsomorphismRepresentatives    Examples:

HAPAAA    Examples:

HAPBARCODE    Examples:

HAPDerivationType    Examples:

HAPPRIME_LastLHSBicomplexSize    Examples:

HAPPRIME_ShuffleRandomSource    Examples:

HAPRIGXXX    Examples:

HAP_GCOMPLEX_LIST    Examples:

HAP_GCOMPLEX_SETUP    Examples:

HAP_MOVES_DIM_2    Examples:

HAP_MOVES_DIM_3    Examples:

HAP_PERMMOVES_DIM_2    Examples:

HAP_PERMMOVES_DIM_3    Examples:

HAP_PoincareCubeManifoldEdgeDegrees    Examples:

HAP_Test    Examples:

HAP_XYXYXYXY    Examples:

HAPchildFunctionToggle    Examples:

HAPchildToggle    Examples:

HAPchildren    Examples:

HapConjQuandElt    Examples:

HapConjQuandEltFamily    Examples:

HapConstantPolRing    Examples:

HapEquivariantChainComplex    Examples:

HapEquivariantChainComplexFamily    Examples:

HapEquivariantNonFreeChainComplex    Examples:

HapEquivariantNonFreeChainComplexFamily    Examples:

HapEquivariantSpectralSequencePage    Examples:

HapEquivariantSpectralSequencePageFamily    Examples:

HapGComplexMap    Examples:

HapGComplexMapFamily    Examples:

HapQuandlePresentation    Examples:

HapQuandlePresentationFamily    Examples:

HapRightTransversalSL2ZSubgroup    Examples:

HapSL2ZConjugatedSubgroup    Examples:

HapSL2ZSubgroup    Examples:

HapSimplicialFreeAbelianGroup    Examples:

HapSimplicialFreeAbelianGroupFamily    Examples:

HapTorsionSubcomplex    Examples:

HapTorsionSubcomplexFamily    Examples:

IntersectionForm    Examples: 1 , 2 

IsHapRightTransversalSL2ZSubgroup    Examples:

IsHapSL2ConjugatedSubgroup    Examples:

IsHapSL2OSubgroup    Examples:

IsHapSL2Subgroup    Examples:

IsHapSL2ZConjugatedSubgroup    Examples:

IsHapSL2ZSubgroup    Examples:

RefinedColouring_gc    Examples:

RefinedColouring_group    Examples:

RegularCWAssociahedronWithDiscreteVectorField    Examples:

RegularCWClosedSurface    Examples:

RegularCWComplexWithAttachedRelatorCells    Examples: 1 

RegularCWComplex_DisjointUnion    Examples:

RegularCWComplex_WedgeSum    Examples:

RegularCWDiscreteSpace    Examples: 1 

RegularCWSphere    Examples: 1 

SimplicialComplexConnectedSum    Examples:

SphericalKnotComplementWithBoundary    Examples:

StemGroups    Examples:

cat    Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 

cnt    Examples:

hap_cr    Examples:

 [Top of Book]  [Contents]   [Previous Chapter]   [Next Chapter] 
diff --git a/doc/chap5.html b/doc/chap5.html index c2a4d9dd..6f646da2 100644 --- a/doc/chap5.html +++ b/doc/chap5.html @@ -318,7 +318,7 @@
5.1-20 ResolutionNilpotentGroup

The contracting homotopy on the ZE-resolution has not yet been fully implemented for infinite groups.

-

Examples: 1 , 2 , 3 , 4 

+

Examples: 1 , 2 , 3 , 4 , 5 

diff --git a/doc/chap5.txt b/doc/chap5.txt index 0cc592a6..86308a10 100644 --- a/doc/chap5.txt +++ b/doc/chap5.txt @@ -388,9 +388,9 @@ The contracting homotopy on the ZE-resolution has not yet been fully implemented for infinite groups. - Examples: 1 (../tutorial/chap11.html) , 2 - (../www/SideLinks/About/aboutCohomologyRings.html) , 3 - (../www/SideLinks/About/aboutRosenbergerMonster.html) , 4 + Examples: 1 (../tutorial/chap6.html) , 2 (../tutorial/chap11.html) , 3 + (../www/SideLinks/About/aboutCohomologyRings.html) , 4 + (../www/SideLinks/About/aboutRosenbergerMonster.html) , 5 (../www/SideLinks/About/aboutExtensions.html)  5.1-21 ResolutionNormalSeries diff --git a/doc/chap5_mj.html b/doc/chap5_mj.html index e4712faf..12074b3d 100644 --- a/doc/chap5_mj.html +++ b/doc/chap5_mj.html @@ -321,7 +321,7 @@
5.1-20 ResolutionNilpotentGroup

The contracting homotopy on the \(ZE\)-resolution has not yet been fully implemented for infinite groups.

-

Examples: 1 , 2 , 3 , 4 

+

Examples: 1 , 2 , 3 , 4 , 5 

diff --git a/doc/chap9.html b/doc/chap9.html index 9da9c175..da302f6f 100644 --- a/doc/chap9.html +++ b/doc/chap9.html @@ -72,7 +72,7 @@
9.1-3 ChevalleyEilenbergComplex

This function was written by Pablo Fernandez Ascariz

-

Examples:

+

Examples: 1 

diff --git a/doc/chap9.txt b/doc/chap9.txt index 2ecb5c11..69799f83 100644 --- a/doc/chap9.txt +++ b/doc/chap9.txt @@ -47,7 +47,7 @@ This function was written by Pablo Fernandez Ascariz - Examples: + Examples: 1 (../tutorial/chap7.html)  9.1-4 LeibnizComplex diff --git a/doc/chap9_mj.html b/doc/chap9_mj.html index baf64b7f..1baf4e5d 100644 --- a/doc/chap9_mj.html +++ b/doc/chap9_mj.html @@ -75,7 +75,7 @@
9.1-3 ChevalleyEilenbergComplex

This function was written by Pablo Fernandez Ascariz

-

Examples:

+

Examples: 1 

diff --git a/doc/chapInd.html b/doc/chapInd.html index 811e79dd..8cce949e 100644 --- a/doc/chapInd.html +++ b/doc/chapInd.html @@ -70,7 +70,7 @@

Index

BoundaryOfPureCubicalComplex 29.1-36
BoundingPureCubicalComplex 29.1-40
CategoricalEnrichment 34.2-1
-CategoryName 34.2-7
+CategoryName 34.2-9
CayleyGraphOfGroup 1.1-7 1.10-3 28.1-18
CayleyGraphOfGroupDisplay 17.1-1 17.1-1
CayleyMetric 1.2-1 33.1-1 33.1-1
@@ -108,7 +108,7 @@

Index

ComplementOfFilteredPureCubicalComplex 29.1-49
ComplementOfPureCubicalComplex 29.1-42
Compose 39.1-5
-CompositionEqualityAdditionMinus 34.2-8
+CompositionEqualityAdditionMinus 34.2-10
CompositionSeriesOfFpGModules 21.1-1
ConcentricFiltration 1.3-3
ConjugatedResolution 5.1-27
@@ -223,6 +223,8 @@

Index

HAPDerivation 2.4-2
HAPPrintTo 36.1-8
HAPRead 36.1-9
+HasInitialObject 34.2-5
+HasTerminalObject 34.2-6
HilbertPoincareSeries 2.4-3
Homology 1.9-3 1.9-3 1.9-3 1.9-3 1.9-3 1.9-3 1.9-3 11.1-18 28.1-1 28.1-1 29.1-15 29.1-15
HomologyOfDerivation 2.4-4
@@ -253,8 +255,8 @@

Index

IntersectionOfFpGModules 21.1-12
IsAspherical 1.5-7 17.1-3
IsAvailableChild 36.1-5
-IsCategoryArrow 34.2-12
-IsCategoryObject 34.2-11
+IsCategoryArrow 34.2-14
+IsCategoryObject 34.2-13
IsConnectedQuandle 32.1-15
IsFpGModuleHomomorphismData 21.1-13
IsLatin 32.1-14
@@ -287,7 +289,7 @@

Index

MakeHAPManual 39.1-9
ManhattanMetric 1.2-6 33.1-6
Map 38.1-7
-Mapping 34.2-10
+Mapping 34.2-12
MaximalSimplicesToSimplicialComplex 28.1-10
MaximalSubmoduleOfFpGModule 21.1-14
MaximalSubmodulesOfFpGModule 21.1-15
@@ -315,7 +317,7 @@

Index

NormalSeriesToQuotientHomomorphisms 39.1-13
NormalSubgroupAsCatOneGroup 24.1-7
NumberOfHomomorphisms 32.1-6
-Object 34.2-9
+Object 34.2-11
OrbitPolytope 1.10-10 18.1-6
OrientRegularCWComplex 1.4-5
ParallelList 37.1-4
@@ -455,7 +457,7 @@

Index

SkeletonOfSimplicialComplex 28.1-11
SL2Z 39.1-1 39.1-1
SolutionsMatDestructive 39.1-11
-Source 34.2-5 38.1-8
+Source 34.2-7 38.1-8
SparseBoundaryMatrix 10.1-12
SparseChainComplex 10.1-10
SparseChainComplexOfRegularCWComplex 10.1-11
@@ -474,7 +476,7 @@

Index

SymmetricMatrixToGraph 1.1-24
SymmetricMatrixToIncidenceMatrix 28.1-16 28.1-16
Syzygy 19.1-4
-Target 34.2-6 38.1-9
+Target 34.2-8 38.1-9
TensorCentre 15.1-12
TensorProductOfChainComplexes 9.1-8
TensorWithIntegers 2.3-3 2.3-3 8.1-9
diff --git a/doc/chapInd.txt b/doc/chapInd.txt index ab9f4e29..e315c5a5 100644 --- a/doc/chapInd.txt +++ b/doc/chapInd.txt @@ -47,7 +47,7 @@ BoundaryOfPureCubicalComplex 29.1-36 BoundingPureCubicalComplex 29.1-40 CategoricalEnrichment 34.2-1 - CategoryName 34.2-7 + CategoryName 34.2-9 CayleyGraphOfGroup 1.1-7 1.10-3 28.1-18 CayleyGraphOfGroupDisplay 17.1-1 17.1-1 CayleyMetric 1.2-1 33.1-1 33.1-1 @@ -85,7 +85,7 @@ ComplementOfFilteredPureCubicalComplex 29.1-49 ComplementOfPureCubicalComplex 29.1-42 Compose 39.1-5 - CompositionEqualityAdditionMinus 34.2-8 + CompositionEqualityAdditionMinus 34.2-10 CompositionSeriesOfFpGModules 21.1-1 ConcentricFiltration 1.3-3 ConjugatedResolution 5.1-27 @@ -200,6 +200,8 @@ HAPDerivation 2.4-2 HAPPrintTo 36.1-8 HAPRead 36.1-9 + HasInitialObject 34.2-5 + HasTerminalObject 34.2-6 HilbertPoincareSeries 2.4-3 Homology 1.9-3 1.9-3 1.9-3 1.9-3 1.9-3 1.9-3 1.9-3 11.1-18 28.1-1 28.1-1 29.1-15 29.1-15 HomologyOfDerivation 2.4-4 @@ -230,8 +232,8 @@ IntersectionOfFpGModules 21.1-12 IsAspherical 1.5-7 17.1-3 IsAvailableChild 36.1-5 - IsCategoryArrow 34.2-12 - IsCategoryObject 34.2-11 + IsCategoryArrow 34.2-14 + IsCategoryObject 34.2-13 IsConnectedQuandle 32.1-15 IsFpGModuleHomomorphismData 21.1-13 IsLatin 32.1-14 @@ -264,7 +266,7 @@ MakeHAPManual 39.1-9 ManhattanMetric 1.2-6 33.1-6 Map 38.1-7 - Mapping 34.2-10 + Mapping 34.2-12 MaximalSimplicesToSimplicialComplex 28.1-10 MaximalSubmoduleOfFpGModule 21.1-14 MaximalSubmodulesOfFpGModule 21.1-15 @@ -292,7 +294,7 @@ NormalSeriesToQuotientHomomorphisms 39.1-13 NormalSubgroupAsCatOneGroup 24.1-7 NumberOfHomomorphisms 32.1-6 - Object 34.2-9 + Object 34.2-11 OrbitPolytope 1.10-10 18.1-6 OrientRegularCWComplex 1.4-5 ParallelList 37.1-4 @@ -432,7 +434,7 @@ SkeletonOfSimplicialComplex 28.1-11 SL2Z 39.1-1 39.1-1 SolutionsMatDestructive 39.1-11 - Source 34.2-5 38.1-8 + Source 34.2-7 38.1-8 SparseBoundaryMatrix 10.1-12 SparseChainComplex 10.1-10 SparseChainComplexOfRegularCWComplex 10.1-11 @@ -451,7 +453,7 @@ SymmetricMatrixToGraph 1.1-24 SymmetricMatrixToIncidenceMatrix 28.1-16 28.1-16 Syzygy 19.1-4 - Target 34.2-6 38.1-9 + Target 34.2-8 38.1-9 TensorCentre 15.1-12 TensorProductOfChainComplexes 9.1-8 TensorWithIntegers 2.3-3 2.3-3 8.1-9 diff --git a/doc/chapInd_mj.html b/doc/chapInd_mj.html index 3965af06..1c102c33 100644 --- a/doc/chapInd_mj.html +++ b/doc/chapInd_mj.html @@ -73,7 +73,7 @@

Index

BoundaryOfPureCubicalComplex 29.1-36
BoundingPureCubicalComplex 29.1-40
CategoricalEnrichment 34.2-1
-CategoryName 34.2-7
+CategoryName 34.2-9
CayleyGraphOfGroup 1.1-7 1.10-3 28.1-18
CayleyGraphOfGroupDisplay 17.1-1 17.1-1
CayleyMetric 1.2-1 33.1-1 33.1-1
@@ -111,7 +111,7 @@

Index

ComplementOfFilteredPureCubicalComplex 29.1-49
ComplementOfPureCubicalComplex 29.1-42
Compose 39.1-5
-CompositionEqualityAdditionMinus 34.2-8
+CompositionEqualityAdditionMinus 34.2-10
CompositionSeriesOfFpGModules 21.1-1
ConcentricFiltration 1.3-3
ConjugatedResolution 5.1-27
@@ -226,6 +226,8 @@

Index

HAPDerivation 2.4-2
HAPPrintTo 36.1-8
HAPRead 36.1-9
+HasInitialObject 34.2-5
+HasTerminalObject 34.2-6
HilbertPoincareSeries 2.4-3
Homology 1.9-3 1.9-3 1.9-3 1.9-3 1.9-3 1.9-3 1.9-3 11.1-18 28.1-1 28.1-1 29.1-15 29.1-15
HomologyOfDerivation 2.4-4
@@ -256,8 +258,8 @@

Index

IntersectionOfFpGModules 21.1-12
IsAspherical 1.5-7 17.1-3
IsAvailableChild 36.1-5
-IsCategoryArrow 34.2-12
-IsCategoryObject 34.2-11
+IsCategoryArrow 34.2-14
+IsCategoryObject 34.2-13
IsConnectedQuandle 32.1-15
IsFpGModuleHomomorphismData 21.1-13
IsLatin 32.1-14
@@ -290,7 +292,7 @@

Index

MakeHAPManual 39.1-9
ManhattanMetric 1.2-6 33.1-6
Map 38.1-7
-Mapping 34.2-10
+Mapping 34.2-12
MaximalSimplicesToSimplicialComplex 28.1-10
MaximalSubmoduleOfFpGModule 21.1-14
MaximalSubmodulesOfFpGModule 21.1-15
@@ -318,7 +320,7 @@

Index

NormalSeriesToQuotientHomomorphisms 39.1-13
NormalSubgroupAsCatOneGroup 24.1-7
NumberOfHomomorphisms 32.1-6
-Object 34.2-9
+Object 34.2-11
OrbitPolytope 1.10-10 18.1-6
OrientRegularCWComplex 1.4-5
ParallelList 37.1-4
@@ -458,7 +460,7 @@

Index

SkeletonOfSimplicialComplex 28.1-11
SL2Z 39.1-1 39.1-1
SolutionsMatDestructive 39.1-11
-Source 34.2-5 38.1-8
+Source 34.2-7 38.1-8
SparseBoundaryMatrix 10.1-12
SparseChainComplex 10.1-10
SparseChainComplexOfRegularCWComplex 10.1-11
@@ -477,7 +479,7 @@

Index

SymmetricMatrixToGraph 1.1-24
SymmetricMatrixToIncidenceMatrix 28.1-16 28.1-16
Syzygy 19.1-4
-Target 34.2-6 38.1-9
+Target 34.2-8 38.1-9
TensorCentre 15.1-12
TensorProductOfChainComplexes 9.1-8
TensorWithIntegers 2.3-3 2.3-3 8.1-9
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\bibcite{eichler}{Eic57} @@ -336,6 +350,7 @@ \bibcite{kuzmin}{KS98} \bibcite{ksontini}{Kso00} \bibcite{kulkarni}{Kul91} +\bibcite{MartinFTM01}{MFTM01} \bibcite{milnor}{Mil58} \bibcite{moise}{Moi52} \bibcite{lmoser}{Mos71} @@ -352,4 +367,4 @@ \bibcite{thurston}{Thu02} \bibcite{tomoda}{TZ08} \bibcite{wieser}{Wie78} -\@writefile{toc}{\contentsline {chapter}{References}{174}{chapter*.2}\protected@file@percent } +\@writefile{toc}{\contentsline {chapter}{References}{181}{chapter*.2}\protected@file@percent } diff --git a/tutorial/HapTutorial.bbl b/tutorial/HapTutorial.bbl index 8c269e30..c74893bd 100644 --- a/tutorial/HapTutorial.bbl +++ b/tutorial/HapTutorial.bbl @@ -1,4 +1,4 @@ -\begin{thebibliography}{BCNS15} +\begin{thebibliography}{MFTM01} \bibitem[AL70]{atkinlehner} A.~Atkin and J.~Lehner. @@ -32,6 +32,12 @@ E.~Brody. \newblock {\em The topological classification of the lens spaces}. \newblock Ann. of Math. 71, 163{\textendash}184, 1960. +\bibitem[CKL14]{coeurjolly} +D.~Coeurjolly, B.~Kerautret, and J.-O. Lachaud. +\newblock {\em Extraction of Connected Region Boundary in Multidimensional + Images}. +\newblock Image Processing On Line, 2014. + \bibitem[DPR91]{dpr} R.~Dijkgraaf, V.~Pasquier, and P.~Roche. \newblock {\em Quasi-Hopf algebras, group cohomology and orbifold models}. @@ -106,6 +112,12 @@ R.~Kulkarni. modular group. \newblock {\em American Journal of Mathematics}, 113, No. 6:1053--1133, 1991. +\bibitem[MFTM01]{MartinFTM01} +D.~Martin, C.~Fowlkes, D.~Tal, and J.~Malik. +\newblock {\em A Database of Human Segmented Natural Images and its Application + to Evaluating Segmentation Algorithms and Measuring Ecological Statistics}. +\newblock Proc. 8th Int'l Conf. Computer Vision, 2, pp 416--423, 2001. + \bibitem[Mil58]{milnor} J.~Milnor. \newblock {\em On simply connected 4-manifolds}. diff --git a/tutorial/HapTutorial.blg b/tutorial/HapTutorial.blg index 3efd8a84..d5ac0023 100644 --- a/tutorial/HapTutorial.blg +++ b/tutorial/HapTutorial.blg @@ -3,44 +3,44 @@ Capacity: max_strings=200000, hash_size=200000, hash_prime=170003 The top-level auxiliary file: HapTutorial.aux The style file: alpha.bst Database file #1: mybib.xml.bib -You've used 36 entries, +You've used 38 entries, 2543 wiz_defined-function locations, - 752 strings with 8829 characters, -and the built_in function-call counts, 11632 in all, are: -= -- 1100 -> -- 455 + 761 strings with 9245 characters, +and the built_in function-call counts, 12404 in all, are: += -- 1167 +> -- 511 < -- 24 -+ -- 142 -- -- 136 -* -- 713 -:= -- 1960 -add.period$ -- 108 -call.type$ -- 36 -change.case$ -- 173 -chr.to.int$ -- 36 -cite$ -- 36 -duplicate$ -- 551 -empty$ -- 952 -format.name$ -- 195 -if$ -- 2347 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{brownloday}{{132}{12.6}{section.12.6}} -\backcite {eichler}{{139}{13.1}{section.13.1}} -\backcite {shimura}{{139}{13.1}{section.13.1}} -\backcite {stein}{{140}{13.1}{section.13.1}} -\backcite {wieser}{{140}{13.1}{section.13.1}} -\backcite {ellisharrisskoldberg}{{141}{13.3}{section.13.3}} -\backcite {kulkarni}{{142}{13.3}{section.13.3}} -\backcite {buiellis}{{142}{13.4}{section.13.4}} -\backcite {stein}{{146}{13.6}{section.13.6}} -\backcite {stein}{{147}{13.8}{section.13.8}} -\backcite {atkinlehner}{{147}{13.8}{section.13.8}} -\backcite {swan}{{151}{13.10}{section.13.10}} -\backcite {rahmthesis}{{151}{13.10}{section.13.10}} -\backcite {schoennenbeck}{{151}{13.10}{section.13.10}} -\backcite {sengun}{{152}{13.10}{section.13.10}} -\backcite {bergeron}{{157}{13.14}{section.13.14}} +\backcite {ksontini}{{9}{1.4}{section.1.4}} +\backcite {spreerkhuenel}{{10}{1.6}{section.1.6}} +\backcite {spreerkhuenel}{{15}{1.10}{section.1.10}} +\backcite {milnor}{{18}{1.11}{section.1.11}} +\backcite {goncalves}{{19}{1.12}{section.1.12}} +\backcite {MR1758871}{{35}{3.2}{section.3.2}} +\backcite {MR2441256}{{35}{3.2}{section.3.2}} +\backcite {reidemeister}{{46}{4.6}{section.4.6}} +\backcite {moise}{{46}{4.6}{section.4.6}} +\backcite {brody}{{46}{4.6}{section.4.6}} +\backcite {przytycki}{{46}{4.6}{section.4.6}} +\backcite {lmoser}{{49}{4.7}{section.4.7}} +\backcite {greene}{{49}{4.7}{section.4.7}} +\backcite {thurston}{{62}{4.14}{section.4.14}} +\backcite {MartinFTM01}{{70}{5.6}{section.5.6}} +\backcite {coeurjolly}{{71}{5.7}{section.5.7}} +\backcite {coeurjolly}{{71}{5.7}{section.5.7}} +\backcite {kuzmin}{{92}{7.7}{section.7.7}} +\backcite {igusa}{{92}{7.7}{section.7.7}} +\backcite {tomoda}{{99}{7.12}{section.7.12}} +\backcite {dpr}{{100}{7.13}{section.7.13}} +\backcite {horadam}{{100}{7.13}{section.7.13}} +\backcite {swan2}{{122}{11.2}{section.11.2}} +\backcite {kholodna}{{122}{11.2}{section.11.2}} +\backcite {johnson}{{122}{11.2}{section.11.2}} +\backcite {hatcher}{{137}{12.4}{section.12.4}} +\backcite {brownloday}{{139}{12.6}{section.12.6}} +\backcite {eichler}{{146}{13.1}{section.13.1}} +\backcite {shimura}{{146}{13.1}{section.13.1}} +\backcite {stein}{{147}{13.1}{section.13.1}} +\backcite {wieser}{{147}{13.1}{section.13.1}} +\backcite {ellisharrisskoldberg}{{148}{13.3}{section.13.3}} +\backcite {kulkarni}{{149}{13.3}{section.13.3}} +\backcite {buiellis}{{149}{13.4}{section.13.4}} +\backcite {stein}{{153}{13.6}{section.13.6}} +\backcite {stein}{{154}{13.8}{section.13.8}} +\backcite {atkinlehner}{{154}{13.8}{section.13.8}} +\backcite {swan}{{158}{13.10}{section.13.10}} +\backcite {rahmthesis}{{158}{13.10}{section.13.10}} +\backcite {schoennenbeck}{{158}{13.10}{section.13.10}} +\backcite {sengun}{{159}{13.10}{section.13.10}} +\backcite {bergeron}{{164}{13.14}{section.13.14}} diff --git a/tutorial/HapTutorial.log b/tutorial/HapTutorial.log index c1a522a1..befea48d 100644 --- a/tutorial/HapTutorial.log +++ b/tutorial/HapTutorial.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.14159265-2.6-1.40.20 (TeX Live 2019/Debian) (preloaded format=pdflatex 2023.5.31) 5 JAN 2024 22:22 +This is pdfTeX, Version 3.14159265-2.6-1.40.20 (TeX Live 2019/Debian) (preloaded format=pdflatex 2023.5.31) 1 FEB 2024 12:24 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -435,12 +435,12 @@ LaTeX Font Info: Font shape `OT1/ptm/bx/n' in size <6> not available (Font) Font shape `OT1/ptm/b/n' tried instead on input line 2. [2 -] [3] [4]) +] [3] [4] [5]) \tf@toc=\write7 \openout7 = `HapTutorial.toc'. -[5] +[6] Chapter 1. LaTeX Font Info: Trying to load font information for T1+cmtt on input line 1 34. @@ -449,10 +449,10 @@ File: t1cmtt.fd 2019/12/16 v2.5j Standard LaTeX font definitions ) LaTeX Font Info: Font shape `T1/cmtt/bx/n' in size <10> not available (Font) Font shape `T1/cmtt/m/n' tried instead on input line 135. - [6 + [7 -] [7] [8] [9] -[10] [11] [12] [13] [14] [15] +] [8] [9] [10] +[11] [12] [13] [14] [15] [16] LaTeX Font Info: Font shape `OT1/ptm/bx/n' in size <14.4> not available (Font) Font shape `OT1/ptm/b/n' tried instead on input line 713. @@ -464,47 +464,47 @@ Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 713. -[16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] +[17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] Chapter 2. -[27 +[28 ] LaTeX Font Info: Trying to load font information for TS1+ptm on input line 1 340. (/usr/share/texlive/texmf-dist/tex/latex/psnfss/ts1ptm.fd File: ts1ptm.fd 2001/06/04 font definitions for TS1/ptm. -) [28] [29] -[30] +) [29] [30] +[31] Underfull \hbox (badness 10000) in paragraph at lines 1484--1487 [] -[31] +[32] Chapter 3. -[32 +[33 -] [33] [34] [35] [36] +] [34] [35] [36] [37] Underfull \hbox (badness 1668) in paragraph at lines 1817--1818 []\T1/ptm/m/n/10.95 Continuing with the above ex-am-ple where $\OML/ztmcm/m/it/ 10.95 Y$ \T1/ptm/m/n/10.95 is the real pro-jec-tive plane, we see that [] -[37] [38] [39] [40] +[38] [39] [40] [41] Chapter 4. -[41 +[42 -] [42]pdfTeX warning (ext4): destination with the same identifier (name{L.X80B6 +] [43]pdfTeX warning (ext4): destination with the same identifier (name{L.X80B6 849C835B7F19}) has been already used, duplicate ignored \relax l.2114 \hyperdef{L}{X80B6849C835B7F19}{} - [43] [44] + [44] [45] Underfull \hbox (badness 3039) in paragraph at lines 2245--2252 \T1/ptm/m/n/10.95 In 1935 K. Rei-de-meis-ter [[]][] clas-si-fied lens spaces up to ori-en-ta-tion pre-serv-ing PL- [] -[45] [46] [47] +[46] [47] [48] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 2401. @@ -513,248 +513,246 @@ Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 2401. -[48] [49] +[49] [50] LaTeX Font Warning: Font shape `T1/ptm/m/scit' undefined (Font) using `T1/ptm/m/sc' instead on input line 2487. -[50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] +[51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] Underfull \hbox (badness 1675) in paragraph at lines 3078--3083 \T1/ptm/m/n/10.95 topes. A poly-tope of par-tic-u-lar in-ter-est, and one that ap-pears sev-eral times in the clas- [] -[61] [62] [63] +[62] [63] [64] Chapter 5. -[64 +[65 -] [65]pdfTeX warning (ext4): destination with the same identifier (name{L.X7D51 +] [66]pdfTeX warning (ext4): destination with the same identifier (name{L.X7D51 2DA37F789B4C}) has been already used, duplicate ignored \relax -l.3386 \hyperdef{L}{X7D512DA37F789B4C}{} - [66pdfTeX warning (ext4): destination -with the same identifier (name{L.X7D512DA37F789B4C}) has been already used, dup -licate ignored - -\AtBegShi@Output ...ipout \box \AtBeginShipoutBox - \fi \fi -l.3407 ...tsize=\small,frame=single,label=Example] - ] [67] [68] [69] [70] -[71] [72] [73] +l.3332 \hyperdef{L}{X7D512DA37F789B4C}{} + [67] [68]pdfTeX warning (ext4): destin +ation with the same identifier (name{L.X7D512DA37F789B4C}) has been already use +d, duplicate ignored + + \relax +l.3473 \hyperdef{L}{X7D512DA37F789B4C}{} + [69] [70] [71] [72] [73] [74] [75] +[76] [77] [78] Chapter 6. -[74 +[79 ] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `math shift' on input line 3836. +(hyperref) removing `math shift' on input line 4094. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `math shift' on input line 3836. +(hyperref) removing `math shift' on input line 4094. -[75] -Underfull \hbox (badness 3386) in paragraph at lines 3897--3898 +[80] [81] +Underfull \hbox (badness 3386) in paragraph at lines 4233--4234 []\T1/ptm/m/n/10.95 Consider the group $\OML/ztmcm/m/it/10.95 H \OT1/ztmcm/m/n/ 10.95 = \OML/ztmcm/m/it/10.95 SmallGroup\OT1/ztmcm/m/n/10.95 (64\OML/ztmcm/m/it /10.95 ; \OT1/ztmcm/m/n/10.95 134)$\T1/ptm/m/n/10.95 . Con-sider the nor-mal su b-group $\OML/ztmcm/m/it/10.95 N \OT1/ztmcm/m/n/10.95 = [] -[76] [77] [78] +[82] [83] [84] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `math shift' on input line 4033. +(hyperref) removing `math shift' on input line 4369. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `math shift' on input line 4033. +(hyperref) removing `math shift' on input line 4369. -[79] [80] +[85] [86] Chapter 7. -[81 +[87 -] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] -Underfull \hbox (badness 2452) in paragraph at lines 4809--4813 +] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] +Underfull \hbox (badness 2452) in paragraph at lines 5145--5149 \T1/ptm/m/n/10.95 In the oc-ta-he-dral case with $\OML/ztmcm/m/it/10.95 m \OT1/ ztmcm/m/n/10.95 = 1$ \T1/ptm/m/n/10.95 we ob-tain $$H\T1/cmtt/m/n/10.95 ^\\T1/p tm/m/n/10.95 ast(\T1/cmtt/m/n/10.95 \\T1/ptm/m/n/10.95 Gamma,\T1/cmtt/m/n/10.95 \\T1/ptm/m/n/10.95 mathbb Z) = \T1/cmtt/m/n/10.95 \\T1/ptm/m/n/10.95 mathbb [] -[92] -Underfull \hbox (badness 2302) in paragraph at lines 4833--4837 +[98] +Underfull \hbox (badness 2302) in paragraph at lines 5169--5173 \T1/ptm/m/n/10.95 In the tetra-he-dral case with $\OML/ztmcm/m/it/10.95 m \OT1/ ztmcm/m/n/10.95 = 1$ \T1/ptm/m/n/10.95 we ob-tain $$H\T1/cmtt/m/n/10.95 ^\\T1/p tm/m/n/10.95 ast(\T1/cmtt/m/n/10.95 \\T1/ptm/m/n/10.95 Gamma,\T1/cmtt/m/n/10.95 \\T1/ptm/m/n/10.95 mathbb Z) = \T1/cmtt/m/n/10.95 \\T1/ptm/m/n/10.95 mathbb [] -[93] [94] -Underfull \hbox (badness 5217) in paragraph at lines 4926--4927 +[99] [100] +Underfull \hbox (badness 5217) in paragraph at lines 5262--5263 \T1/ptm/m/n/10.95 A non-standard $\OT1/ztmcm/m/n/10.95 3$\T1/ptm/m/n/10.95 -coc ycle $\OML/ztmcm/m/it/10.95 f$ \T1/ptm/m/n/10.95 can be con-verted to a stan-da rd one us-ing the com-mand [] -[95] +[101] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `math shift' on input line 4998. +(hyperref) removing `math shift' on input line 5334. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `math shift' on input line 4998. +(hyperref) removing `math shift' on input line 5334. -[96] [97] [98] +[102] [103] [104] [105] Chapter 8. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `math shift' on input line 5128. +(hyperref) removing `math shift' on input line 5485. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `math shift' on input line 5128. +(hyperref) removing `math shift' on input line 5485. -Underfull \hbox (badness 1975) in paragraph at lines 5131--5133 +Underfull \hbox (badness 1975) in paragraph at lines 5488--5490 \T1/ptm/m/n/10.95 com-putes a fi-nite di-men-sional graded ring equal to the co -ho-mol-ogy ring $\OML/ztmcm/m/it/10.95 H[]\OT1/ztmcm/m/n/10.95 (\OML/ztmcm/m/i t/10.95 G; \U/msb/m/n/10.95 Z[]\OT1/ztmcm/m/n/10.95 ) := [] LaTeX Font Info: Font shape `OT1/ptm/bx/n' in size <12> not available -(Font) Font shape `OT1/ptm/b/n' tried instead on input line 5161. +(Font) Font shape `OT1/ptm/b/n' tried instead on input line 5518. LaTeX Font Info: Font shape `OT1/ptm/bx/n' in size <9> not available -(Font) Font shape `OT1/ptm/b/n' tried instead on input line 5161. +(Font) Font shape `OT1/ptm/b/n' tried instead on input line 5518. LaTeX Font Info: Font shape `OT1/ptm/bx/n' in size <7> not available -(Font) Font shape `OT1/ptm/b/n' tried instead on input line 5161. -[99 +(Font) Font shape `OT1/ptm/b/n' tried instead on input line 5518. +[106 ] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `math shift' on input line 5185. +(hyperref) removing `math shift' on input line 5542. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `math shift' on input line 5185. +(hyperref) removing `math shift' on input line 5542. -[100] [101] +[107] [108] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `math shift' on input line 5310. +(hyperref) removing `math shift' on input line 5667. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `math shift' on input line 5310. +(hyperref) removing `math shift' on input line 5667. 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Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `math shift' on input line 5417. +(hyperref) removing `math shift' on input line 5774. -[104] +[111] Chapter 9. pdfTeX warning (ext4): destination with the same identifier (name{L.X7AFFB32587 D047FE}) has been already used, duplicate ignored \relax -l.5466 \hyperdef{L}{X7AFFB32587D047FE}{} - [105 +l.5823 \hyperdef{L}{X7AFFB32587D047FE}{} + [112 -] [106] +] [113] Chapter 10. -[107 +[114 -] [108] [109] [110] [111] [112] [113] +] [115] [116] [117] [118] [119] [120] Chapter 11. -[114 +[121 -] [115] +] [122] [123] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `math shift' on input line 5993. +(hyperref) removing `math shift' on input line 6374. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `math shift' on input line 5993. +(hyperref) removing `math shift' on input line 6374. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `math shift' on input line 5993. +(hyperref) removing `math shift' on input line 6374. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `subscript' on input line 5993. +(hyperref) removing `subscript' on input line 6374. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `math shift' on input line 5993. +(hyperref) removing `math shift' on input line 6374. -[116] [117] [118] [119] [120] -Underfull \hbox (badness 1715) in paragraph at lines 6258--6261 +[124] [125] [126] [127] +Underfull \hbox (badness 1715) in paragraph at lines 6639--6642 \T1/ptm/m/n/10.95 An im-ple-men-ta-tion of the above method for Bieber-bach gro ups is also avail-able for ar-bi- [] -Underfull \hbox (badness 1152) in paragraph at lines 6258--6261 +Underfull \hbox (badness 1152) in paragraph at lines 6639--6642 \T1/ptm/m/n/10.95 trary crys-tal-lo-graphic groups. The fol-low-ing ex-am-ple c on-structs a res-o-lu-tion for the group [] -[121] +[128] [129] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `math shift' on input line 6350. +(hyperref) removing `math shift' on input line 6731. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `subscript' on input line 6350. +(hyperref) removing `subscript' on input line 6731. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `math shift' on input line 6350. +(hyperref) removing `math shift' on input line 6731. -[122] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `math shift' on input line 6365. +(hyperref) removing `math shift' on input line 6746. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `subscript' on input line 6365. +(hyperref) removing `subscript' on input line 6746. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `\@ifnextchar' on input line 6365. +(hyperref) removing `\@ifnextchar' on input line 6746. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `math shift' on input line 6365. +(hyperref) removing `math shift' on input line 6746. -Underfull \hbox (badness 3260) in paragraph at lines 6368--6371 +Underfull \hbox (badness 3260) in paragraph at lines 6749--6752 \T1/ptm/m/n/10.95 The fol-low-ing uses fi-nite "Voronoi com-plexes" and ho-mo-l og-i-cal per-tur-ba-tion to con-struct [] -Underfull \hbox (badness 2717) in paragraph at lines 6368--6371 +Underfull \hbox (badness 2717) in paragraph at lines 6749--6752 \T1/ptm/m/n/10.95 a res-o-lu-tion for $\OML/ztmcm/m/it/10.95 G \OT1/ztmcm/m/n/1 0.95 = \OML/ztmcm/m/it/10.95 SL[]\OT1/ztmcm/m/n/10.95 (\OMS/ztmcm/m/n/10.95 O\O T1/ztmcm/m/n/10.95 (\U/msb/m/n/10.95 Q\OT1/ztmcm/m/n/10.95 ([]))$\T1/ptm/m/n/10 @@ -762,35 +760,35 @@ T1/ztmcm/m/n/10.95 (\U/msb/m/n/10.95 Q\OT1/ztmcm/m/n/10.95 ([]))$\T1/ptm/m/n/10 [] -Underfull \hbox (badness 1281) in paragraph at lines 6368--6371 +Underfull \hbox (badness 1281) in paragraph at lines 6749--6752 \T1/ptm/m/n/10.95 dently by A. Rahm, M. Dutour-Scikiric and S. Schoe-nen-beck a nd are stored in the folder [] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `math shift' on input line 6383. +(hyperref) removing `math shift' on input line 6764. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `subscript' on input line 6383. +(hyperref) removing `subscript' on input line 6764. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `\@ifnextchar' on input line 6383. +(hyperref) removing `\@ifnextchar' on input line 6764. 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Rahm, M. Dutour-Scikiric and S. Schoe-nen-beck a nd are stored in the folder [] -[123] [124] +[130] [131] [132] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `math shift' on input line 6516. +(hyperref) removing `math shift' on input line 6897. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `math shift' on input line 6516. +(hyperref) removing `math shift' on input line 6897. -[125] [126] +[133] Chapter 12. -[127 +[134 ] -Underfull \hbox (badness 2626) in paragraph at lines 6615--6615 +Underfull \hbox (badness 2626) in paragraph at lines 6996--6996 [][]\T1/ptm/b/n/14.4 Eilenberg-MacLane spaces as sim-pli-cial groups (not rec-o m- [] -[128] [129] +[135] [136] Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `math shift' on input line 6710. +(hyperref) removing `math shift' on input line 7091. 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Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `math shift' on input line 7790. +(hyperref) removing `math shift' on input line 8171. -[147] [148] [149] [150] [151] [152] -Underfull \hbox (badness 10000) in paragraph at lines 8088--8091 +[154] [155] [156] [157] [158] [159] +Underfull \hbox (badness 10000) in paragraph at lines 8469--8472 \T1/ptm/m/n/10.95 can be used to com-pute res-o-lu-tions for groups whose data (pro-vided by Se-bas- [] -Underfull \hbox (badness 5862) in paragraph at lines 8088--8091 +Underfull \hbox (badness 5862) in paragraph at lines 8469--8472 \T1/ptm/m/n/10.95 tian Schoen-nen-beck, Alexan-der Rahm and Math-ieu Du-tour) i s stored in the di-rec-tory [] -[153] [154] [155] -Underfull \hbox (badness 3930) in paragraph at lines 8282--8283 +[160] [161] [162] +Underfull \hbox (badness 3930) in paragraph at lines 8663--8664 []\T1/ptm/m/n/10.95 The next com-mands first con-struct the con-gru-ence sub-gr oup $\OT1/ztmcm/m/n/10.95 ^^@[](\OML/ztmcm/m/it/10.95 I\OT1/ztmcm/m/n/10.95 )$ \T1/ptm/m/n/10.95 of in-dex $\OT1/ztmcm/m/n/10.95 144$ \T1/ptm/m/n/10.95 in [] -[156] [157] [158] +[163] [164] [165] Chapter 14. -[159 +[166 -] [160] +] [167] Chapter 15. -[161 +[168 -] [162] [163] [164] [165] [166] [167] [168] [169] [170] (./HapTutorial.bbl +] [169] [170] [171] [172] [173] [174] [175] [176] [177] (./HapTutorial.bbl (./HapTutorial.brf) \tf@brf=\write8 \openout8 = `HapTutorial.brf'. - [171] [172 + [178] [179 -] [173] -Underfull \hbox (badness 10000) in paragraph at lines 183--183 +] [180] +Underfull \hbox (badness 10000) in paragraph at lines 195--195 []\T1/ptm/m/n/10.95 W. 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segmentation}{chapter.5}% 54 -\BOOKMARK [1][-]{section.5.5}{Alternative approaches to computing persistent homology}{chapter.5}% 55 -\BOOKMARK [1][-]{section.5.6}{Knotted proteins}{chapter.5}% 56 -\BOOKMARK [1][-]{section.5.7}{Random simplicial complexes}{chapter.5}% 57 -\BOOKMARK [1][-]{section.5.8}{Computing homology of a clique complex \(Vietoris-Rips complex\) }{chapter.5}% 58 -\BOOKMARK [0][-]{chapter.6}{Group theoretic computations}{}% 59 -\BOOKMARK [1][-]{section.6.1}{Third homotopy group of a supsension of an Eilenberg-MacLane space }{chapter.6}% 60 -\BOOKMARK [1][-]{section.6.2}{Representations of knot quandles}{chapter.6}% 61 -\BOOKMARK [1][-]{section.6.3}{Identifying knots}{chapter.6}% 62 -\BOOKMARK [1][-]{section.6.4}{Aspherical 2-complexes}{chapter.6}% 63 -\BOOKMARK [1][-]{section.6.5}{Bogomolov multiplier}{chapter.6}% 64 -\BOOKMARK [1][-]{section.6.6}{Second group cohomology and group extensions}{chapter.6}% 65 -\BOOKMARK [1][-]{section.6.7}{Second group cohomology and cocyclic Hadamard matrices}{chapter.6}% 66 -\BOOKMARK [1][-]{section.6.8}{Third group cohomology and homotopy 2-types}{chapter.6}% 67 -\BOOKMARK [0][-]{chapter.7}{Cohomology of groups}{}% 68 -\BOOKMARK [1][-]{section.7.1}{Finite groups }{chapter.7}% 69 -\BOOKMARK [1][-]{section.7.2}{Nilpotent groups}{chapter.7}% 70 -\BOOKMARK [1][-]{section.7.3}{Crystallographic and Almost Crystallographic groups}{chapter.7}% 71 -\BOOKMARK [1][-]{section.7.4}{Arithmetic groups}{chapter.7}% 72 -\BOOKMARK [1][-]{section.7.5}{Artin groups}{chapter.7}% 73 -\BOOKMARK [1][-]{section.7.6}{Graphs of groups}{chapter.7}% 74 -\BOOKMARK [1][-]{section.7.7}{Lie algebra homology and free nilpotent groups}{chapter.7}% 75 -\BOOKMARK [1][-]{section.7.8}{Cohomology with coefficients in a module}{chapter.7}% 76 -\BOOKMARK [1][-]{section.7.9}{Cohomology as a functor of the first variable}{chapter.7}% 77 -\BOOKMARK [1][-]{section.7.10}{Cohomology as a functor of the second variable and the long exact coefficient sequence}{chapter.7}% 78 -\BOOKMARK [1][-]{section.7.11}{Transfer Homomorphism}{chapter.7}% 79 -\BOOKMARK [1][-]{section.7.12}{Cohomology rings of finite fundamental groups of 3-manifolds }{chapter.7}% 80 -\BOOKMARK [1][-]{section.7.13}{Explicit cocycles }{chapter.7}% 81 -\BOOKMARK [1][-]{section.7.14}{Quillen's complex and the p-part of homology }{chapter.7}% 82 -\BOOKMARK [0][-]{chapter.8}{Cohomology rings and Steenrod operations for finite groups}{}% 83 -\BOOKMARK [1][-]{section.8.1}{Mod-p cohomology rings of finite groups}{chapter.8}% 84 -\BOOKMARK [1][-]{section.8.2}{Functorial ring homomorphisms in Mod-p cohomology}{chapter.8}% 85 -\BOOKMARK [1][-]{section.8.3}{Cohomology rings of finite 2-groups}{chapter.8}% 86 -\BOOKMARK [1][-]{section.8.4}{Steenrod operations for finite 2-groups}{chapter.8}% 87 -\BOOKMARK [1][-]{section.8.5}{Steenrod operations on the classifying space of a finite p-group}{chapter.8}% 88 -\BOOKMARK [0][-]{chapter.9}{Bredon homology}{}% 89 -\BOOKMARK [1][-]{section.9.1}{Davis complex}{chapter.9}% 90 -\BOOKMARK [1][-]{section.9.2}{Arithmetic groups}{chapter.9}% 91 -\BOOKMARK [1][-]{section.9.3}{Crystallographic groups}{chapter.9}% 92 -\BOOKMARK [0][-]{chapter.10}{Chain Complexes}{}% 93 -\BOOKMARK [1][-]{section.10.1}{Chain complex of a simplicial complex and simplicial pair}{chapter.10}% 94 -\BOOKMARK [1][-]{section.10.2}{Chain complex of a cubical complex and cubical pair}{chapter.10}% 95 -\BOOKMARK [1][-]{section.10.3}{Chain complex of a regular CW-complex}{chapter.10}% 96 -\BOOKMARK [1][-]{section.10.4}{Chain Maps of simplicial and regular CW maps}{chapter.10}% 97 -\BOOKMARK [1][-]{section.10.5}{Constructions for chain complexes}{chapter.10}% 98 -\BOOKMARK [1][-]{section.10.6}{Filtered chain complexes}{chapter.10}% 99 -\BOOKMARK [1][-]{section.10.7}{Sparse chain complexes}{chapter.10}% 100 -\BOOKMARK [0][-]{chapter.11}{Resolutions}{}% 101 -\BOOKMARK [1][-]{section.11.1}{Resolutions for small finite groups}{chapter.11}% 102 -\BOOKMARK [1][-]{section.11.2}{Resolutions for very small finite groups}{chapter.11}% 103 -\BOOKMARK [1][-]{section.11.3}{Resolutions for finite groups acting on orbit polytopes}{chapter.11}% 104 -\BOOKMARK [1][-]{section.11.4}{Minimal resolutions for finite p-groups over Fp}{chapter.11}% 105 -\BOOKMARK [1][-]{section.11.5}{Resolutions for abelian groups}{chapter.11}% 106 -\BOOKMARK [1][-]{section.11.6}{Resolutions for nilpotent groups}{chapter.11}% 107 -\BOOKMARK [1][-]{section.11.7}{Resolutions for groups with subnormal series}{chapter.11}% 108 -\BOOKMARK [1][-]{section.11.8}{Resolutions for groups with normal series}{chapter.11}% 109 -\BOOKMARK [1][-]{section.11.9}{Resolutions for polycyclic \(almost\) crystallographic groups }{chapter.11}% 110 -\BOOKMARK [1][-]{section.11.10}{Resolutions for Bieberbach groups }{chapter.11}% 111 -\BOOKMARK [1][-]{section.11.11}{Resolutions for arbitrary crystallographic groups}{chapter.11}% 112 -\BOOKMARK [1][-]{section.11.12}{Resolutions for crystallographic groups admitting cubical fundamental domain}{chapter.11}% 113 -\BOOKMARK [1][-]{section.11.13}{Resolutions for Coxeter groups }{chapter.11}% 114 -\BOOKMARK [1][-]{section.11.14}{Resolutions for Artin groups }{chapter.11}% 115 -\BOOKMARK [1][-]{section.11.15}{Resolutions for G=SL2\(Z[1/m]\)}{chapter.11}% 116 -\BOOKMARK [1][-]{section.11.16}{Resolutions for selected groups G=SL2\( O\(Q\(d\) \)}{chapter.11}% 117 -\BOOKMARK [1][-]{section.11.17}{Resolutions for selected groups G=PSL2\( O\(Q\(d\) \)}{chapter.11}% 118 -\BOOKMARK [1][-]{section.11.18}{Resolutions for a few higher-dimensional arithmetic groups }{chapter.11}% 119 -\BOOKMARK [1][-]{section.11.19}{Resolutions for finite-index subgroups }{chapter.11}% 120 -\BOOKMARK [1][-]{section.11.20}{Simplifying resolutions }{chapter.11}% 121 -\BOOKMARK [1][-]{section.11.21}{Resolutions for graphs of groups and for groups with aspherical presentations }{chapter.11}% 122 -\BOOKMARK [1][-]{section.11.22}{Resolutions for FG-modules }{chapter.11}% 123 -\BOOKMARK [0][-]{chapter.12}{Simplicial groups}{}% 124 -\BOOKMARK [1][-]{section.12.1}{Crossed modules}{chapter.12}% 125 -\BOOKMARK [1][-]{section.12.2}{Eilenberg-MacLane spaces as simplicial groups \(not recommended\)}{chapter.12}% 126 -\BOOKMARK [1][-]{section.12.3}{Eilenberg-MacLane spaces as simplicial free abelian groups \(recommended\)}{chapter.12}% 127 -\BOOKMARK [1][-]{section.12.4}{Elementary theoretical information on H\(K\(,n\),Z\)}{chapter.12}% 128 -\BOOKMARK [1][-]{section.12.5}{The first three non-trivial homotopy groups of spheres}{chapter.12}% 129 -\BOOKMARK [1][-]{section.12.6}{The first two non-trivial homotopy groups of the suspension and double suspension of a K\(G,1\)}{chapter.12}% 130 -\BOOKMARK [1][-]{section.12.7}{Postnikov towers and 5\(S3\) }{chapter.12}% 131 -\BOOKMARK [1][-]{section.12.8}{Towards 4\(K\(G,1\)\) }{chapter.12}% 132 -\BOOKMARK [1][-]{section.12.9}{Enumerating homotopy 2-types}{chapter.12}% 133 -\BOOKMARK [1][-]{section.12.10}{Identifying cat1-groups of low order}{chapter.12}% 134 -\BOOKMARK [1][-]{section.12.11}{Identifying crossed modules of low order}{chapter.12}% 135 -\BOOKMARK [0][-]{chapter.13}{Congruence Subgroups, Cuspidal Cohomology and Hecke Operators}{}% 136 -\BOOKMARK [1][-]{section.13.1}{Eichler-Shimura isomorphism}{chapter.13}% 137 -\BOOKMARK [1][-]{section.13.2}{Generators for SL2\(Z\) and the cubic tree}{chapter.13}% 138 -\BOOKMARK [1][-]{section.13.3}{One-dimensional fundamental domains and generators for congruence subgroups}{chapter.13}% 139 -\BOOKMARK [1][-]{section.13.4}{Cohomology of congruence subgroups}{chapter.13}% 140 -\BOOKMARK [1][-]{section.13.5}{Cuspidal cohomology}{chapter.13}% 141 -\BOOKMARK [1][-]{section.13.6}{Hecke operators on forms of weight 2}{chapter.13}% 142 -\BOOKMARK [1][-]{section.13.7}{Hecke operators on forms of weight \0402}{chapter.13}% 143 -\BOOKMARK [1][-]{section.13.8}{Reconstructing modular forms from cohomology computations}{chapter.13}% 144 -\BOOKMARK [1][-]{section.13.9}{The Picard group}{chapter.13}% 145 -\BOOKMARK [1][-]{section.13.10}{Bianchi groups}{chapter.13}% 146 -\BOOKMARK [1][-]{section.13.11}{Some other infinite matrix groups}{chapter.13}% 147 -\BOOKMARK [1][-]{section.13.12}{Ideals and finite quotient groups}{chapter.13}% 148 -\BOOKMARK [1][-]{section.13.13}{Congruence subgroups for ideals}{chapter.13}% 149 -\BOOKMARK [1][-]{section.13.14}{First homology}{chapter.13}% 150 -\BOOKMARK [0][-]{chapter.14}{Parallel computation}{}% 151 -\BOOKMARK [1][-]{section.14.1}{An embarassingly parallel computation}{chapter.14}% 152 -\BOOKMARK [1][-]{section.14.2}{An non-embarassingly parallel computation}{chapter.14}% 153 -\BOOKMARK [0][-]{chapter.15}{Regular CW-structure on knots \(written by Kelvin Killeen\)}{}% 154 -\BOOKMARK [1][-]{section.15.1}{Knot complements in the 3-ball}{chapter.15}% 155 -\BOOKMARK [1][-]{section.15.2}{Tubular neighbourhoods}{chapter.15}% 156 -\BOOKMARK [1][-]{section.15.3}{Knotted surface complements in the 4-ball}{chapter.15}% 157 -\BOOKMARK [0][-]{chapter*.2}{References}{}% 158 +\BOOKMARK [1][-]{section.5.3}{Some tools for handling pure complexes}{chapter.5}% 53 +\BOOKMARK [1][-]{section.5.4}{Digital image analysis and persistent homology}{chapter.5}% 54 +\BOOKMARK [1][-]{section.5.5}{A second example of digital image segmentation}{chapter.5}% 55 +\BOOKMARK [1][-]{section.5.6}{A third example of digital image segmentation}{chapter.5}% 56 +\BOOKMARK [1][-]{section.5.7}{Naive example of digital image contour extraction}{chapter.5}% 57 +\BOOKMARK [1][-]{section.5.8}{Alternative approaches to computing persistent homology}{chapter.5}% 58 +\BOOKMARK [1][-]{section.5.9}{Knotted proteins}{chapter.5}% 59 +\BOOKMARK [1][-]{section.5.10}{Random simplicial complexes}{chapter.5}% 60 +\BOOKMARK [1][-]{section.5.11}{Computing homology of a clique complex \(Vietoris-Rips complex\) }{chapter.5}% 61 +\BOOKMARK [0][-]{chapter.6}{Group theoretic computations}{}% 62 +\BOOKMARK [1][-]{section.6.1}{Third homotopy group of a supsension of an Eilenberg-MacLane space }{chapter.6}% 63 +\BOOKMARK [1][-]{section.6.2}{Representations of knot quandles}{chapter.6}% 64 +\BOOKMARK [1][-]{section.6.3}{Identifying knots}{chapter.6}% 65 +\BOOKMARK [1][-]{section.6.4}{Aspherical 2-complexes}{chapter.6}% 66 +\BOOKMARK [1][-]{section.6.5}{Group presentations and homotopical syzygies}{chapter.6}% 67 +\BOOKMARK [1][-]{section.6.6}{Bogomolov multiplier}{chapter.6}% 68 +\BOOKMARK [1][-]{section.6.7}{Second group cohomology and group extensions}{chapter.6}% 69 +\BOOKMARK [1][-]{section.6.8}{Second group cohomology and cocyclic Hadamard matrices}{chapter.6}% 70 +\BOOKMARK [1][-]{section.6.9}{Third group cohomology and homotopy 2-types}{chapter.6}% 71 +\BOOKMARK [0][-]{chapter.7}{Cohomology of groups \(and Lie Algebras\)}{}% 72 +\BOOKMARK [1][-]{section.7.1}{Finite groups }{chapter.7}% 73 +\BOOKMARK [1][-]{section.7.2}{Nilpotent groups}{chapter.7}% 74 +\BOOKMARK [1][-]{section.7.3}{Crystallographic and Almost Crystallographic groups}{chapter.7}% 75 +\BOOKMARK [1][-]{section.7.4}{Arithmetic groups}{chapter.7}% 76 +\BOOKMARK [1][-]{section.7.5}{Artin groups}{chapter.7}% 77 +\BOOKMARK [1][-]{section.7.6}{Graphs of groups}{chapter.7}% 78 +\BOOKMARK [1][-]{section.7.7}{Lie algebra homology and free nilpotent groups}{chapter.7}% 79 +\BOOKMARK [1][-]{section.7.8}{Cohomology with coefficients in a module}{chapter.7}% 80 +\BOOKMARK [1][-]{section.7.9}{Cohomology as a functor of the first variable}{chapter.7}% 81 +\BOOKMARK [1][-]{section.7.10}{Cohomology as a functor of the second variable and the long exact coefficient sequence}{chapter.7}% 82 +\BOOKMARK [1][-]{section.7.11}{Transfer Homomorphism}{chapter.7}% 83 +\BOOKMARK [1][-]{section.7.12}{Cohomology rings of finite fundamental groups of 3-manifolds }{chapter.7}% 84 +\BOOKMARK [1][-]{section.7.13}{Explicit cocycles }{chapter.7}% 85 +\BOOKMARK [1][-]{section.7.14}{Quillen's complex and the p-part of homology }{chapter.7}% 86 +\BOOKMARK [1][-]{section.7.15}{Homology of a Lie algebra with coefficients in a module}{chapter.7}% 87 +\BOOKMARK [0][-]{chapter.8}{Cohomology rings and Steenrod operations for finite groups}{}% 88 +\BOOKMARK [1][-]{section.8.1}{Mod-p cohomology rings of finite groups}{chapter.8}% 89 +\BOOKMARK [1][-]{section.8.2}{Functorial ring homomorphisms in Mod-p cohomology}{chapter.8}% 90 +\BOOKMARK [1][-]{section.8.3}{Cohomology rings of finite 2-groups}{chapter.8}% 91 +\BOOKMARK [1][-]{section.8.4}{Steenrod operations for finite 2-groups}{chapter.8}% 92 +\BOOKMARK [1][-]{section.8.5}{Steenrod operations on the classifying space of a finite p-group}{chapter.8}% 93 +\BOOKMARK [0][-]{chapter.9}{Bredon homology}{}% 94 +\BOOKMARK [1][-]{section.9.1}{Davis complex}{chapter.9}% 95 +\BOOKMARK [1][-]{section.9.2}{Arithmetic groups}{chapter.9}% 96 +\BOOKMARK [1][-]{section.9.3}{Crystallographic groups}{chapter.9}% 97 +\BOOKMARK [0][-]{chapter.10}{Chain Complexes}{}% 98 +\BOOKMARK [1][-]{section.10.1}{Chain complex of a simplicial complex and simplicial pair}{chapter.10}% 99 +\BOOKMARK [1][-]{section.10.2}{Chain complex of a cubical complex and cubical pair}{chapter.10}% 100 +\BOOKMARK [1][-]{section.10.3}{Chain complex of a regular CW-complex}{chapter.10}% 101 +\BOOKMARK [1][-]{section.10.4}{Chain Maps of simplicial and regular CW maps}{chapter.10}% 102 +\BOOKMARK [1][-]{section.10.5}{Constructions for chain complexes}{chapter.10}% 103 +\BOOKMARK [1][-]{section.10.6}{Filtered chain complexes}{chapter.10}% 104 +\BOOKMARK [1][-]{section.10.7}{Sparse chain complexes}{chapter.10}% 105 +\BOOKMARK [0][-]{chapter.11}{Resolutions}{}% 106 +\BOOKMARK [1][-]{section.11.1}{Resolutions for small finite groups}{chapter.11}% 107 +\BOOKMARK [1][-]{section.11.2}{Resolutions for very small finite groups}{chapter.11}% 108 +\BOOKMARK [1][-]{section.11.3}{Resolutions for finite groups acting on orbit polytopes}{chapter.11}% 109 +\BOOKMARK [1][-]{section.11.4}{Minimal resolutions for finite p-groups over Fp}{chapter.11}% 110 +\BOOKMARK [1][-]{section.11.5}{Resolutions for abelian groups}{chapter.11}% 111 +\BOOKMARK [1][-]{section.11.6}{Resolutions for nilpotent groups}{chapter.11}% 112 +\BOOKMARK [1][-]{section.11.7}{Resolutions for groups with subnormal series}{chapter.11}% 113 +\BOOKMARK [1][-]{section.11.8}{Resolutions for groups with normal series}{chapter.11}% 114 +\BOOKMARK [1][-]{section.11.9}{Resolutions for polycyclic \(almost\) crystallographic groups }{chapter.11}% 115 +\BOOKMARK [1][-]{section.11.10}{Resolutions for Bieberbach groups }{chapter.11}% 116 +\BOOKMARK [1][-]{section.11.11}{Resolutions for arbitrary crystallographic groups}{chapter.11}% 117 +\BOOKMARK [1][-]{section.11.12}{Resolutions for crystallographic groups admitting cubical fundamental domain}{chapter.11}% 118 +\BOOKMARK [1][-]{section.11.13}{Resolutions for Coxeter groups }{chapter.11}% 119 +\BOOKMARK [1][-]{section.11.14}{Resolutions for Artin groups }{chapter.11}% 120 +\BOOKMARK [1][-]{section.11.15}{Resolutions for G=SL2\(Z[1/m]\)}{chapter.11}% 121 +\BOOKMARK [1][-]{section.11.16}{Resolutions for selected groups G=SL2\( O\(Q\(d\) \)}{chapter.11}% 122 +\BOOKMARK [1][-]{section.11.17}{Resolutions for selected groups G=PSL2\( O\(Q\(d\) \)}{chapter.11}% 123 +\BOOKMARK [1][-]{section.11.18}{Resolutions for a few higher-dimensional arithmetic groups }{chapter.11}% 124 +\BOOKMARK [1][-]{section.11.19}{Resolutions for finite-index subgroups }{chapter.11}% 125 +\BOOKMARK [1][-]{section.11.20}{Simplifying resolutions }{chapter.11}% 126 +\BOOKMARK [1][-]{section.11.21}{Resolutions for graphs of groups and for groups with aspherical presentations }{chapter.11}% 127 +\BOOKMARK [1][-]{section.11.22}{Resolutions for FG-modules }{chapter.11}% 128 +\BOOKMARK [0][-]{chapter.12}{Simplicial groups}{}% 129 +\BOOKMARK [1][-]{section.12.1}{Crossed modules}{chapter.12}% 130 +\BOOKMARK [1][-]{section.12.2}{Eilenberg-MacLane spaces as simplicial groups \(not recommended\)}{chapter.12}% 131 +\BOOKMARK [1][-]{section.12.3}{Eilenberg-MacLane spaces as simplicial free abelian groups \(recommended\)}{chapter.12}% 132 +\BOOKMARK [1][-]{section.12.4}{Elementary theoretical information on H\(K\(,n\),Z\)}{chapter.12}% 133 +\BOOKMARK [1][-]{section.12.5}{The first three non-trivial homotopy groups of spheres}{chapter.12}% 134 +\BOOKMARK [1][-]{section.12.6}{The first two non-trivial homotopy groups of the suspension and double suspension of a K\(G,1\)}{chapter.12}% 135 +\BOOKMARK [1][-]{section.12.7}{Postnikov towers and 5\(S3\) }{chapter.12}% 136 +\BOOKMARK [1][-]{section.12.8}{Towards 4\(K\(G,1\)\) }{chapter.12}% 137 +\BOOKMARK [1][-]{section.12.9}{Enumerating homotopy 2-types}{chapter.12}% 138 +\BOOKMARK [1][-]{section.12.10}{Identifying cat1-groups of low order}{chapter.12}% 139 +\BOOKMARK [1][-]{section.12.11}{Identifying crossed modules of low order}{chapter.12}% 140 +\BOOKMARK [0][-]{chapter.13}{Congruence Subgroups, Cuspidal Cohomology and Hecke Operators}{}% 141 +\BOOKMARK [1][-]{section.13.1}{Eichler-Shimura isomorphism}{chapter.13}% 142 +\BOOKMARK [1][-]{section.13.2}{Generators for SL2\(Z\) and the cubic tree}{chapter.13}% 143 +\BOOKMARK [1][-]{section.13.3}{One-dimensional fundamental domains and generators for congruence subgroups}{chapter.13}% 144 +\BOOKMARK [1][-]{section.13.4}{Cohomology of congruence subgroups}{chapter.13}% 145 +\BOOKMARK [1][-]{section.13.5}{Cuspidal cohomology}{chapter.13}% 146 +\BOOKMARK [1][-]{section.13.6}{Hecke operators on forms of weight 2}{chapter.13}% 147 +\BOOKMARK [1][-]{section.13.7}{Hecke operators on forms of weight \0402}{chapter.13}% 148 +\BOOKMARK [1][-]{section.13.8}{Reconstructing modular forms from cohomology computations}{chapter.13}% 149 +\BOOKMARK [1][-]{section.13.9}{The Picard group}{chapter.13}% 150 +\BOOKMARK [1][-]{section.13.10}{Bianchi groups}{chapter.13}% 151 +\BOOKMARK [1][-]{section.13.11}{Some other infinite matrix groups}{chapter.13}% 152 +\BOOKMARK [1][-]{section.13.12}{Ideals and finite quotient groups}{chapter.13}% 153 +\BOOKMARK [1][-]{section.13.13}{Congruence subgroups for ideals}{chapter.13}% 154 +\BOOKMARK [1][-]{section.13.14}{First homology}{chapter.13}% 155 +\BOOKMARK [0][-]{chapter.14}{Parallel computation}{}% 156 +\BOOKMARK [1][-]{section.14.1}{An embarassingly parallel computation}{chapter.14}% 157 +\BOOKMARK [1][-]{section.14.2}{An non-embarassingly parallel computation}{chapter.14}% 158 +\BOOKMARK [0][-]{chapter.15}{Regular CW-structure on knots \(written by Kelvin Killeen\)}{}% 159 +\BOOKMARK [1][-]{section.15.1}{Knot complements in the 3-ball}{chapter.15}% 160 +\BOOKMARK [1][-]{section.15.2}{Tubular neighbourhoods}{chapter.15}% 161 +\BOOKMARK [1][-]{section.15.3}{Knotted surface complements in the 4-ball}{chapter.15}% 162 +\BOOKMARK [0][-]{chapter*.2}{References}{}% 163 diff --git a/tutorial/HapTutorial.pnr b/tutorial/HapTutorial.pnr index 36618511..42967bac 100644 --- a/tutorial/HapTutorial.pnr +++ b/tutorial/HapTutorial.pnr @@ -1,180 +1,187 @@ PAGENRS := [ [ 0, 0, 0 ], 1, [ 0, 0, 1 ], 2, -[ 1, 0, 0 ], 6, -[ 1, 1, 0 ], 6, -[ 1, 2, 0 ], 7, -[ 1, 3, 0 ], 7, -[ 1, 4, 0 ], 8, -[ 1, 5, 0 ], 8, -[ 1, 6, 0 ], 9, -[ 1, 7, 0 ], 9, -[ 1, 8, 0 ], 10, -[ 1, 9, 0 ], 11, -[ 1, 10, 0 ], 12, -[ 1, 11, 0 ], 17, -[ 1, 12, 0 ], 18, -[ 1, 13, 0 ], 19, -[ 1, 14, 0 ], 20, -[ 1, 15, 0 ], 20, -[ 1, 16, 0 ], 21, -[ 1, 17, 0 ], 23, -[ 2, 0, 0 ], 27, -[ 2, 1, 0 ], 27, -[ 2, 2, 0 ], 28, -[ 2, 3, 0 ], 30, -[ 2, 4, 0 ], 31, -[ 3, 0, 0 ], 32, -[ 3, 1, 0 ], 32, -[ 3, 2, 0 ], 33, -[ 3, 3, 0 ], 35, -[ 3, 4, 0 ], 36, -[ 3, 5, 0 ], 36, -[ 3, 6, 0 ], 37, -[ 3, 6, 1 ], 37, -[ 3, 6, 2 ], 38, -[ 3, 7, 0 ], 39, -[ 4, 0, 0 ], 41, -[ 4, 1, 0 ], 41, -[ 4, 2, 0 ], 42, -[ 4, 3, 0 ], 42, -[ 4, 4, 0 ], 43, -[ 4, 5, 0 ], 44, -[ 4, 6, 0 ], 45, -[ 4, 7, 0 ], 47, -[ 4, 8, 0 ], 48, -[ 4, 9, 0 ], 49, -[ 4, 10, 0 ], 50, -[ 4, 10, 1 ], 52, -[ 4, 10, 2 ], 56, -[ 4, 11, 0 ], 56, -[ 4, 12, 0 ], 58, -[ 4, 13, 0 ], 60, -[ 4, 14, 0 ], 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b/tutorial/HapTutorial.tex index e44f8e10..93024a26 100644 --- a/tutorial/HapTutorial.tex +++ b/tutorial/HapTutorial.tex @@ -90,12 +90,12 @@ \begin{titlepage} \mbox{}\vfill -\begin{center}{\maintitlesize \textbf{A short HAP tutorial\mbox{}}}\\ +\begin{center}{\maintitlesize \textbf{A newer HAP tutorial\mbox{}}}\\ \vfill -\hypersetup{pdftitle=A short HAP tutorial} -\markright{\scriptsize \mbox{}\hfill A short HAP tutorial \hfill\mbox{}} -{\Huge \textbf{(\href{../www/SideLinks/About/aboutContents.html} {A more comprehensive tutorial is available here}\\ +\hypersetup{pdftitle=A newer HAP tutorial} +\markright{\scriptsize \mbox{}\hfill A newer HAP tutorial \hfill\mbox{}} +{\Huge \textbf{(\href{../www/SideLinks/About/aboutContents.html} {An older tutorial is available here}\\ and\\ \href{https://global.oup.com/academic/product/an-invitation-to-computational-homotopy-9780198832980} {A related book is available here}\\ and\\ @@ -3245,6 +3245,13 @@ \section{\textcolor{Chapter }{Persistent homology }}\logpage{[ 5, 1, 0 ]} \end{Verbatim} +The first 54 terms in the filtration each have 74 path components -- one for +each point in the sample. During the next 9 filtration terms the number of +path components reduces, meaning that sample points begin to coalesce due to +the formation of edges in the simplicial complexes. Then, two path components +persist over an interval of 18 filtration terms, before they eventually +coalesce. + The next commands display the resulting degree $1$ persistent homology as a barcode. \begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] !gapprompt@gap>| !gapinput@P:=PersistentBettiNumbers(K,1);;| @@ -3253,7 +3260,9 @@ \section{\textcolor{Chapter }{Persistent homology }}\logpage{[ 5, 1, 0 ]} \end{Verbatim} - The following command displays the $1$ skeleton of the simplicial complex arizing as the $65$-th term in the filtration on the clique complex. + Interpreting short bars as noise, we see for instance that the $65$th term in the filtration could be regarded as noiseless and belonging to a +"stable interval" in the filtration with regards to first and second homology +functors. The following command displays (up to homotopy) the $1$ skeleton of the simplicial complex arizing as the $65$-th term in the filtration on the clique complex. \begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] !gapprompt@gap>| !gapinput@Y:=FiltrationTerm(K,65);| Regular CW-complex of dimension 1 @@ -3263,7 +3272,7 @@ \section{\textcolor{Chapter }{Persistent homology }}\logpage{[ 5, 1, 0 ]} \end{Verbatim} -These computations suuggest that the dataset contains two persistent path +These computations suggest that the dataset contains two persistent path components (or clusters), and that each path component is in some sense periodic. The final command displays one possible representation of the data as points on two circles. @@ -3272,7 +3281,7 @@ \subsection{\textcolor{Chapter }{Background to the data}}\logpage{[ 5, 1, 1 ]} { -Each point in the dataset was an image consisting of $732\times 761$ pixels. This point was regarded as a vector in $\mathbb R^{732\times 761}$ and the matrix $D$ was constructed using the Euclidean metric. The images were the following: +Each point in the dataset was an image consisting of $732\times 761$ pixels. This point was regarded as a vector in $\mathbb R^{557052}=\mathbb R^{732\times 761}$ and the matrix $D$ was constructed using the Euclidean metric. The images were the following: } @@ -3284,16 +3293,14 @@ \section{\textcolor{Chapter }{Mapper clustering}}\logpage{[ 5, 2, 0 ]} { -The following example reads in a set $S$ of vectors of rational numbers. It uses the Euclidean distance $d(u,v)$ between vectors. It fixes some vector -\$u{\textunderscore}0\texttt{\symbol{92}}in S\$ and uses the associated -function $f\colon D\rightarrow [0,b] \subset \mathbb R, v\mapsto d(u_0,v)$. In addition, it uses an open cover of the interval $[0,b]$ consisting of $100$ uniformly distributed overlapping open subintervals of radius $r=29$. It also uses a simple clustering algorithm implemented in the function \texttt{cluster}. +The following example reads in a set $S$ of vectors of rational numbers. It uses the Euclidean distance $d(u,v)$ between vectors. It fixes some vector $u_0\in S $ and uses the associated function $f\colon D\rightarrow [0,b] \subset \mathbb R, v\mapsto d(u_0,v)$. In addition, it uses an open cover of the interval $[0,b]$ consisting of $100$ uniformly distributed overlapping open subintervals of radius $r=29$. It also uses a simple clustering algorithm implemented in the function \texttt{cluster}. These ingredients are input into the Mapper clustering procedure to produce a simplicial complex $M$ which is intended to be a representation of the data. The complex $M$ is $1$-dimensional and the final command uses GraphViz software to visualize the graph. The nodes of this simplicial complex are "buckets" containing data points. A data point may reside in several buckets. The number of points in the bucket determines the size of the node. Two nodes are connected by an edge -when their end-point nodes contain common data points. +when they contain common data points. \begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] !gapprompt@gap>| !gapinput@file:=HapFile("data134.txt");;| !gapprompt@gap>| !gapinput@Read(file);| @@ -3333,7 +3340,56 @@ \subsection{\textcolor{Chapter }{Background to the data}}\label{pointcloud} } -\section{\textcolor{Chapter }{Digital image analysis and persistent homology}}\logpage{[ 5, 3, 0 ]} +\section{\textcolor{Chapter }{Some tools for handling pure complexes}}\logpage{[ 5, 3, 0 ]} +\hyperdef{L}{X7BBDE0567DB8C5DA}{} +{ + A CW-complex $X$ is said to be \emph{pure} if all of its top-dimensional cells have a common dimension. There are +instances where such a space $X$ provides a convenient ambient space whose subspaces can be used to model +experimental data. For instance, the plane $X=\mathbb R^2$ admits a pure regular CW-structure whose $2$-cells are open unit squares with integer coordinate vertices. An alternative, +and sometimes preferrable, pure regular CW-structure on $\mathbb R^2$ is one where the $2$-cells are all reguar hexagons with sides of unit length. Any digital image +can be thresholded to produce a black-white image and this black-white image +can naturally be regared as a finite pure cellular subcomplex of either of the +two proposed CW-structures on $\mathbb R^2$. Analogously, thresholding can be used to represent $3$-dimensional greyscale images as finite pure cellular subspaces of cubical or +permutahedral CW-structures on $\mathbb R^3$, and to represent RGB colour photographs as analogous subcomplexes of $\mathbb R^5$. + + In this section we list a few functions for performing basic operations on $n$-dimensional pure cubical and pure permutahedral finite subcomplexes $M$ of $X=R^n$. We refer to $M$ simply as a \emph{pure complex}. In subsequent sections we demonstrate how these few functions on pure +complexes allow for in-depth analysis of experimental data. + +(\textsc{Aside.} The basic operations could equally well be implemented for other +CW-decompositions of $X=\mathbb R^n$ such as the regular CW-decompositions arising as the tessellations by a +fundamental domain of a Bieberbach group (=torsion free crytallographic +group). Moreover, the basic operations could also be implemented for other +manifolds such as an $n$-torus $X=S^1\times S^1 \times \cdots \times S^1$ or $n$-sphere $X=S^n$ or for $X$ the universal cover of some interesting hyperbolic $3$-manifold. An example use of the ambient manifold $X=S^1\times S^1\times S^1$ could be for the construction of a cellular subspace recording the time of +day, day of week and week of the year of crimes committed in a population.) + +\textsc{Basic operations returning pure complexes.} ( Function descriptions available \href{../doc/chap1_mj.html#X7FD50DF6782F00A0} {here}.) +\begin{itemize} +\item \texttt{PureCubicalComplex(binary array)} +\item \texttt{PurePermutahedralComplex(binary array)} +\item \texttt{ReadImageAsPureCubicalComplex(file,threshold)} +\item \texttt{ReadImageSquenceAsPureCubicalComplex(file,threshold)} +\item \texttt{PureComplexBoundary(M)} +\item \texttt{PureComplexComplement(M)} +\item \texttt{PureComplexRandomCell(M)} +\item \texttt{PureComplexThickened(M)} +\item \texttt{ContractedComplex(M, optional subcomplex of M)} +\item \texttt{ExpandedComplex(M, optional supercomplex of M)} +\item \texttt{PureComplexUnion(M,N)} +\item \texttt{PureComplexIntersection(M,N)} +\item \texttt{PureComplexDifference(M,N)} +\item \texttt{FiltrationTerm(F,n)} +\end{itemize} + + +\textsc{Basic operations returning filtered pure complexes.} +\begin{itemize} +\item \texttt{PureComplexThickeningFiltration(M,length)} +\item \texttt{ReadImageAsFilteredPureCubicalComplex(file,length)} +\end{itemize} + } + + +\section{\textcolor{Chapter }{Digital image analysis and persistent homology}}\logpage{[ 5, 4, 0 ]} \hyperdef{L}{X79616D12822FDB9A}{} { @@ -3344,7 +3400,7 @@ \section{\textcolor{Chapter }{Digital image analysis and persistent homology}}\l this filtered complex is calculated in degrees $0$ and $1$ and displayed as two barcodes. \begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] !gapprompt@gap>| !gapinput@file:=HapFile("image1.3.2.png");;| - !gapprompt@gap>| !gapinput@F:=ReadImageAsFilteredPureCubicalComplex(file,20);| + !gapprompt@gap>| !gapinput@F:=ReadImageAsFilteredPureCubicalComplex(file,40);| Filtered pure cubical complex of dimension 2. !gapprompt@gap>| !gapinput@P:=PersistentBettiNumbers(F,0);;| !gapprompt@gap>| !gapinput@BarCodeCompactDisplay(P);| @@ -3358,18 +3414,21 @@ \section{\textcolor{Chapter }{Digital image analysis and persistent homology}}\l \end{Verbatim} -The $20$ persistent bars in the degree $0$ barcode suggest that the image has $20$ objects. The degree $1$ barcode suggests that $14$ (or possibly $17$) of these objects have holes in them. -\subsection{\textcolor{Chapter }{Naive example of image segmentation by automatic thresholding}}\logpage{[ 5, 3, 1 ]} +The $20$ persistent bars in the degree $0$ barcode suggest that the image has $20$ objects. The degree $1$ barcode suggests that there are $14$ (or possibly $17$) holes in these $20$ objects. +\subsection{\textcolor{Chapter }{Naive example of image segmentation by automatic thresholding}}\logpage{[ 5, 4, 1 ]} \hyperdef{L}{X8066F9B17B78418E}{} { Assuming that short bars and isolated points in the barcodes represent noise while long bars represent essential features, a "noiseless" representation of the image should correspond to a term in the filtration corresponding to a -column in the barcode incident with all long bars but incident with no short -bars or isolated points. The following commands confirm that the 4th term in -the filtration is such a term and display this term as a binary image. -\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] - !gapprompt@gap>| !gapinput@Y:=FiltrationTerm(F,4); | +column in the barcode incident with all the long bars but incident with no +short bars or isolated points. There is no noiseless term in the above +filtration of length 40. However (in conjunction with the next subsection) the +following commands confirm that the 64th term in the filtration of length 500 +is such a term and display this term as a binary image. +\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] + !gapprompt@gap>| !gapinput@F:=ReadImageAsFilteredPureCubicalComplex(file,500);;| + !gapprompt@gap>| !gapinput@Y:=FiltrationTerm(F,64); | Pure cubical complex of dimension 2. !gapprompt@gap>| !gapinput@BettiNumber(Y,0);| 20 @@ -3377,12 +3436,40 @@ \subsection{\textcolor{Chapter }{Naive example of image segmentation by automati 14 !gapprompt@gap>| !gapinput@Display(Y);| +\end{Verbatim} + } + + +\subsection{\textcolor{Chapter }{Refining the filtration}}\logpage{[ 5, 4, 2 ]} +\hyperdef{L}{X7E6436E0856761F2}{} +{ + The first filtration for the image has 40 terms. One may wish to investigate a +filtration with more terms, say 500 terms, with a view to analysing, say, +those 1-cycles that are born by term 25 of the filtration and that die between +terms 50 and 60. The following commands produce the relevant barcode showing +that there is precisely one such 1-cycle. +\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] + !gapprompt@gap>| !gapinput@F:=ReadImageAsFilteredPureCubicalComplex(file,500);;| + !gapprompt@gap>| !gapinput@L:=[20,60,61,62,63,64,65,66,67,68,69,70];; | + !gapprompt@gap>| !gapinput@T:=FiltrationTerms(F,L);;| + !gapprompt@gap>| !gapinput@P0:=PersistentBettiNumbers(T,0);;| + !gapprompt@gap>| !gapinput@BarCodeCompactDisplay(P0);| + !gapprompt@gap>| !gapinput@P1:=PersistentBettiNumbers(T,1);;| + !gapprompt@gap>| !gapinput@BarCodeCompactDisplay(P1);| \end{Verbatim} + + +$\beta_0$: + + + + $\beta_1$: + } -\subsection{\textcolor{Chapter }{Background to the data}}\logpage{[ 5, 3, 2 ]} +\subsection{\textcolor{Chapter }{Background to the data}}\logpage{[ 5, 4, 3 ]} \hyperdef{L}{X7D512DA37F789B4C}{} { @@ -3394,19 +3481,19 @@ \subsection{\textcolor{Chapter }{Background to the data}}\logpage{[ 5, 3, 2 ]} } -\section{\textcolor{Chapter }{A second example of digital image segmentation}}\logpage{[ 5, 4, 0 ]} +\section{\textcolor{Chapter }{A second example of digital image segmentation}}\logpage{[ 5, 5, 0 ]} \hyperdef{L}{X7A8224DA7B00E0D9}{} { In order to automatically count the number of coins in this picture - we can load the image as a filtered pure cubical complex $F$ of filtration length 30 say, and observe the degree zero persistent Betti -numbers to establish that the 21-st term or so of $F$ seems to be 'noise free' in degree zero. We can then set $M$ equal to the 21-st term of $F$ and thicken $M$ a couple of times say to remove any tiny holes it may have. We can then + we can load the image as a filtered pure cubical complex $F$ of filtration length 40 say, and observe the degree zero persistent Betti +numbers to establish that the 28-th term or so of $F$ seems to be 'noise free' in degree zero. We can then set $M$ equal to the 28-th term of $F$ and thicken $M$ a couple of times say to remove any tiny holes it may have. We can then construct the complement $C$ of $M$. Then we can construct a 'neighbourhood thickening' filtration $T$ of $C$ with say $50$ consecutive thickenings. The degree one persistent barcode for $T$ has $24$ long bars, suggesting that the original picture consists of $24$ coins. \begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] - !gapprompt@gap>| !gapinput@F:=ReadImageAsFilteredPureCubicalComplex("my_coins.png",30);;| - !gapprompt@gap>| !gapinput@M:=FiltrationTerm(F,21);; #Chosen after viewing degree 0 barcode for F| + !gapprompt@gap>| !gapinput@F:=ReadImageAsFilteredPureCubicalComplex("my_coins.png",40);;| + !gapprompt@gap>| !gapinput@M:=FiltrationTerm(F,24);; #Chosen after viewing degree 0 barcode for F| !gapprompt@gap>| !gapinput@M:=PureComplexThickened(M);;| !gapprompt@gap>| !gapinput@M:=PureComplexThickened(M);;| !gapprompt@gap>| !gapinput@C:=PureComplexComplement(M);;| @@ -3419,11 +3506,11 @@ \section{\textcolor{Chapter }{A second example of digital image segmentation}}\l - The pure cubical complex $M$ has the correct number of path components, namely $25$, but its path components are very much subsets of the regions in the image -corresponding to coins. The complex $M$ can be thickened repeatedly, subject to no two path components being allowed + The pure cubical complex \texttt{W:=PureComplexComplement(FiltrationTerm(T,25))} has the correct number of path components, namely $25$, but its path components are very much subsets of the regions in the image +corresponding to coins. The complex $W$ can be thickened repeatedly, subject to no two path components being allowed to merge, in order to obtain a more realistic image segmentation with path components corresponding more closely to coins. This is done in the follow -commands which use a makeshift function \texttt{Basins(L)} available \href{tutex/basins.g} {here}. The commands essentially implement the standard watershed segmentation +commands which use a makeshift function \texttt{Basins(L)} available \href{tutex/basins.g} {here}. The commands essentially implement a standard watershed segmentation algorithm but do so by using the language of filtered pure cubical complexes. \begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] !gapprompt@gap>| !gapinput@W:=PureComplexComplement(FiltrationTerm(T,25));;| @@ -3443,7 +3530,140 @@ \section{\textcolor{Chapter }{A second example of digital image segmentation}}\l } -\section{\textcolor{Chapter }{Alternative approaches to computing persistent homology}}\logpage{[ 5, 5, 0 ]} +\section{\textcolor{Chapter }{A third example of digital image segmentation}}\logpage{[ 5, 6, 0 ]} +\hyperdef{L}{X8290E7D287F69B98}{} +{ + The following image is number 3096 in the \href{https://www2.eecs.berkeley.edu/Research/Projects/CS/vision/bsds/} {BSDS500 database of images} \cite{MartinFTM01}. + + + +A common first step in segmenting such an image is to appropriately threshold +the corresponding gradient image. + + + + The following commands use the thresholded gradient image to produce an +outline of the aeroplane. The outline is a pure cubical complex with one path +component and with first Betti number equal to 1. +\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] + !gapprompt@gap>| !gapinput@file:=Filename(DirectoriesPackageLibrary("HAP"),"../tutorial/images/3096b.jpg");;| + !gapprompt@gap>| !gapinput@F:=ReadImageAsFilteredPureCubicalComplex(file,30);;| + !gapprompt@gap>| !gapinput@F:=ComplementOfFilteredPureCubicalComplex(F);;| + !gapprompt@gap>| !gapinput@M:=FiltrationTerm(F,27);; #Thickening chosen based on degree 0 barcode| + !gapprompt@gap>| !gapinput@Display(M);;| + !gapprompt@gap>| !gapinput@P:=List([1..BettiNumber(M,0)],n->PathComponentOfPureComplex(M,n));;| + !gapprompt@gap>| !gapinput@P:=Filtered(P,m->Size(m)>10);;| + !gapprompt@gap>| !gapinput@M:=P[1];;| + !gapprompt@gap>| !gapinput@for m in P do| + !gapprompt@>| !gapinput@M:=PureComplexUnion(M,m);;| + !gapprompt@>| !gapinput@od;| + !gapprompt@gap>| !gapinput@T:=ThickeningFiltration(M,50);;| + !gapprompt@gap>| !gapinput@BettiNumber(FiltrationTerm(T,11),0);| + 1 + !gapprompt@gap>| !gapinput@BettiNumber(FiltrationTerm(T,11),1);| + 1 + !gapprompt@gap>| !gapinput@BettiNumber(FiltrationTerm(T,12),1);| + 0 + !gapprompt@gap>| !gapinput@#Confirmation that 11-th filtration term has one hole and the 12-th term is contractible.| + !gapprompt@gap>| !gapinput@C:=FiltrationTerm(T,11);;| + !gapprompt@gap>| !gapinput@for n in Reversed([1..10]) do| + !gapprompt@>| !gapinput@C:=ContractedComplex(C,FiltrationTerm(T,n));| + !gapprompt@>| !gapinput@od;| + !gapprompt@gap>| !gapinput@C:=PureComplexBoundary(PureComplexThickened(C));;| + !gapprompt@gap>| !gapinput@H:=HomotopyEquivalentMinimalPureCubicalSubcomplex(FiltrationTerm(T,12),C);;| + !gapprompt@gap>| !gapinput@B:=ContractedComplex(PureComplexBoundary(H));;| + !gapprompt@gap>| !gapinput@Display(B);| + +\end{Verbatim} + } + + +\section{\textcolor{Chapter }{Naive example of digital image contour extraction}}\logpage{[ 5, 7, 0 ]} +\hyperdef{L}{X7957F329835373E9}{} +{ + The following greyscale image is available from the \href{http://www.ipol.im/pub/art/2014/74/FrechetAndConnectedCompDemo.tgz} {online appendix} to the paper \cite{coeurjolly}. + + + +The following commands produce a picture of contours from this image based on +greyscale values. They also produce a picture of just the closed contours (the +non-closed contours having been homotopy collapsed to points). +\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] + !gapprompt@gap>| !gapinput@file:=Filename(DirectoriesPackageLibrary("HAP"),"../tutorial/images/circularGradient.png");;| + !gapprompt@gap>| !gapinput@L:=[];; | + !gapprompt@gap>| !gapinput@for n in [1..15] do| + !gapprompt@>| !gapinput@M:=ReadImageAsPureCubicalComplex(file,n*30000);| + !gapprompt@>| !gapinput@M:=PureComplexBoundary(M);;| + !gapprompt@>| !gapinput@Add(L,M);| + !gapprompt@>| !gapinput@od;;| + !gapprompt@gap>| !gapinput@C:=L[1];;| + !gapprompt@gap>| !gapinput@for n in [2..Length(L)] do C:=PureComplexUnion(C,L[n]); od;| + !gapprompt@gap>| !gapinput@Display(C);| + !gapprompt@gap>| !gapinput@Display(ContractedComplex(C));| + +\end{Verbatim} + Contours from the above greyscale image: + + + + Closed contours from the above greyscale image: + + + + Very similar results are obtained when applied to the file \texttt{circularGradientNoise.png}, containing noise, available from the \href{http://www.ipol.im/pub/art/2014/74/FrechetAndConnectedCompDemo.tgz} {online appendix} to the paper \cite{coeurjolly}. + +The number of distinct "light sources" in the image can be read from the +countours. Alternatively, this number can be read directly from the barcode +produced by the following commands. +\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] + !gapprompt@gap>| !gapinput@F:=ReadImageAsFilteredPureCubicalComplex(file,20);;| + !gapprompt@gap>| !gapinput@P:=PersistentBettiNumbersAlt(F,1);;| + !gapprompt@gap>| !gapinput@BarCodeCompactDisplay(P);| + +\end{Verbatim} + + + + + The seventeen bars in the barcode correspond to seventeen light sources. The +length of a bar is a measure of the "persistence" of the corresponding light +source. A long bar may initially represent a cluster of several lights whose +members may eventually be distinguished from each other as new bars (or +persistent homology classes) are created. + +Here the command \texttt{PersistentBettiNumbersAlt} has been used. This command is explained in the following section. + +The follwowing commands use a watershed method to partition the digital image +into regions, one region per light source. A makeshift function \texttt{Basins(L)}, available \href{tutex/basins.g} {here}, is called. (The efficiency of the example could be easily improved. For +simplicity it uses generic commands which, in principle, can be applied to +cubical or permutarhedral complexes of higher dimensions.) +\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] + !gapprompt@gap>| !gapinput@file:=Filename(DirectoriesPackageLibrary("HAP"),"../tutorial/images/circularGradient.png");;| + !gapprompt@gap>| !gapinput@F:=ReadImageAsFilteredPureCubicalComplex(file,20);;| + !gapprompt@gap>| !gapinput@FF:=ComplementOfFilteredPureCubicalComplex(F);| + + !gapprompt@gap>| !gapinput@W:=(FiltrationTerm(FF,3));| + !gapprompt@gap>| !gapinput@for n in [4..23] do| + !gapprompt@>| !gapinput@L:=[];;| + !gapprompt@>| !gapinput@for i in [1..PathComponentOfPureComplex(W,0)] do| + !gapprompt@>| !gapinput@ P:=PathComponentOfPureComplex(W,i);;| + !gapprompt@>| !gapinput@ Q:=ThickeningFiltration(P,150,FiltrationTerm(FF,n));;| + !gapprompt@>| !gapinput@ Add(L,Q);;| + !gapprompt@>| !gapinput@od;;| + !gapprompt@>| !gapinput@W:=Basins(L);| + !gapprompt@>| !gapinput@od;| + + !gapprompt@gap>| !gapinput@C:=PureComplexComplement(W);;| + !gapprompt@gap>| !gapinput@T:=PureComplexThickened(C);; C:=ContractedComplex(T,C);; | + !gapprompt@gap>| !gapinput@Display(C);| + +\end{Verbatim} + + + } + + +\section{\textcolor{Chapter }{Alternative approaches to computing persistent homology}}\logpage{[ 5, 8, 0 ]} \hyperdef{L}{X7D2CC9CB85DF1BAF}{} { From any sequence $X_0 \subset X_1 \subset X_2 \subset \cdots \subset X_T$ of cellular spaces (such as pure cubical complexes, or cubical complexes, or @@ -3470,8 +3690,12 @@ \section{\textcolor{Chapter }{Alternative approaches to computing persistent hom In this approach the vector fields on the various spaces $X_k$ are completely independent and so the method lends itself to a degree of easy parallelism. This is not incorporated into the current implementation. - As an illustration we consider a synthetic data set $S$ consisting of $3527$ points sampled, with errors, from an `unknown' manifold $M$ in $\mathbb R^3$. From such a data set one can associate a $3$-dimensional cubical complex $X_0$ consisting of one unit cube centred on each (suitably scaled) data point. -Given a pure cubical complex $X_s$ we construct $X_{s+1} =X_s \cup \{\overline e^3_\lambda\}_{\lambda\in \Lambda}$ by adding to $X_s$ each closed unit cube $\overline e^3_\lambda$ in $\mathbb R^3$ that intersects non-trivially with $X_s$. We construct the filtered cubical complex $X_\ast =\{X_i\}_{0\le i\le 19}$ and compute the persistence matrices $\beta_d^{\ast\ast}$ for $d=0,1,2$ and for $\mathbb Z_2$ coefficients. The filtered complex $X_\ast$ is quite large. In particular, the final space $X_{19}$ in the filtration involves $1\,092727$ vertices, $3\,246354$ edges, $3\,214836$ faces of dimension $2$ and $1\,061208$ faces of dimension $3$. The usual matrix reduction approach to computing persistent Betti numbers + As an illustration we consider a synthetic data set $S$ consisting of $3527$ points sampled, with errors, from an `unknown' manifold $M$ in $\mathbb R^3$. From such a data set one can associate a $3$-dimensional cubical complex $X_0$ consisting of one unit cube centred on each (suitably scaled) data point. A +visualization of $X_0$ is shown below. + + + + Given a pure cubical complex $X_s$ we construct $X_{s+1} =X_s \cup \{\overline e^3_\lambda\}_{\lambda\in \Lambda}$ by adding to $X_s$ each closed unit cube $\overline e^3_\lambda$ in $\mathbb R^3$ that intersects non-trivially with $X_s$. We construct the filtered cubical complex $X_\ast =\{X_i\}_{0\le i\le 19}$ and compute the persistence matrices $\beta_d^{\ast\ast}$ for $d=0,1,2$ and for $\mathbb Z_2$ coefficients. The filtered complex $X_\ast$ is quite large. In particular, the final space $X_{19}$ in the filtration involves $1\,092727$ vertices, $3\,246354$ edges, $3\,214836$ faces of dimension $2$ and $1\,061208$ faces of dimension $3$. The usual matrix reduction approach to computing persistent Betti numbers would involve an appropriate row reduction of sparse matrices one of which has over 3 million rows and 3 million columns. \begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] @@ -3490,7 +3714,7 @@ \section{\textcolor{Chapter }{Alternative approaches to computing persistent hom The barcodes suggest that the data points might have been sampled from a manifold with the homotopy type of a torus. -\subsection{\textcolor{Chapter }{Non-trivial cup product}}\logpage{[ 5, 5, 1 ]} +\subsection{\textcolor{Chapter }{Non-trivial cup product}}\logpage{[ 5, 8, 1 ]} \hyperdef{L}{X86FD0A867EC9E64F}{} { Of course, a wedge $S^2\vee S^1\vee S^1$ has the same homology as the torus $S^1\times S^1$. By establishing that a 'noise free' model for our data points, say the 10-th @@ -3505,10 +3729,44 @@ \subsection{\textcolor{Chapter }{Non-trivial cup product}}\logpage{[ 5, 5, 1 ]} \end{Verbatim} } + +\subsection{\textcolor{Chapter }{Explicit homology generators}}\logpage{[ 5, 8, 2 ]} +\hyperdef{L}{X783EF0F17B629C46}{} +{ + It could be desirable to obtain explicit representatives of the persistent +homology generators that "persist" through a significant sequence of +filtration terms. There are two such generators in degree $1$ and one such generator in degree $2$. The explicit representatives in degree $n$ could consist of an inclusion of pure cubical complexes $Y_n \subset X_{10}$ for which the incuced homology homomorphism $H_n(Y_n,\mathbb Z) \rightarrow H_n(X_{10},\mathbb Z)$ is an isomorphism, and for which $Y_n$ is minimal in the sense that its homotopy type changes if any one or more of +its top dimensional cells are removed. Ideally the space $Y_n$ should be "close to the original dataset" $X_0$. The following commands first construct an explicit degree $2$ homology generator representative $Y_2\subset X_{10}$ where $Y_2$ is homotopy equivalent to $X_{10}$. They then construct an explicit degree $1$ homology generators representative $Y_1\subset X_{10}$ where $Y_1$ is homotopy equivalent to a wedge of two circles. The final command displays +the homology generators representative $Y_1$. +\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] + !gapprompt@gap>| !gapinput@Y2:=FiltrationTerm(F,10);; | + !gapprompt@gap>| !gapinput@for t in Reversed([1..9]) do| + !gapprompt@>| !gapinput@Y2:=ContractedComplex(Y2,FiltrationTerm(F,t));| + !gapprompt@>| !gapinput@od;| + !gapprompt@gap>| !gapinput@Y2:=ContractedComplex(Y2);;| + + !gapprompt@gap>| !gapinput@Size(FiltrationTerm(F,10));| + 918881 + !gapprompt@gap>| !gapinput@Size(Y2); | + 61618 + + !gapprompt@gap>| !gapinput@Y1:=PureComplexDifference(Y2,PureComplexRandomCell(Y2));;| + !gapprompt@gap>| !gapinput@Y1:=ContractedComplex(Y1);;| + !gapprompt@gap>| !gapinput@Size(Y1);| + 474 + !gapprompt@gap>| !gapinput@Display(Y1);| + +\end{Verbatim} + + + + + } + } -\section{\textcolor{Chapter }{Knotted proteins}}\logpage{[ 5, 6, 0 ]} +\section{\textcolor{Chapter }{Knotted proteins}}\logpage{[ 5, 9, 0 ]} \hyperdef{L}{X80D0D8EB7BCD05E9}{} { The \href{https://www.rcsb.org/} {Protein Data Bank} contains a wealth of data which can be investigated with respect to @@ -3532,7 +3790,7 @@ \section{\textcolor{Chapter }{Knotted proteins}}\logpage{[ 5, 6, 0 ]} \end{Verbatim} - + The next command reads in the pdb file for the T.thermophilus 1V2X protein and represents it as a $3$-dimensional pure cubical complex $K$. A resolution of $r=5$ is chosen and this results in a representation as a subcomplex $K$ of an ambient rectangular box of volume equal to $184\times 186\times 294$ unit cubes. The complex $K$ should have the homotopy type of a circle and the protein backbone is a @@ -3549,7 +3807,7 @@ \section{\textcolor{Chapter }{Knotted proteins}}\logpage{[ 5, 6, 0 ]} \end{Verbatim} - + Next we create a filtered pure cubical complex by repeatedly thickening $K$. We perform $15$ thickenings, each thickening being a term in the filtration. The $\beta_1$ barcode for the filtration is displayed. This barcode is a descriptor for the geometry of the protein. For current purposes it suffices to note that the @@ -3587,7 +3845,7 @@ \section{\textcolor{Chapter }{Knotted proteins}}\logpage{[ 5, 6, 0 ]} } -\section{\textcolor{Chapter }{Random simplicial complexes}}\logpage{[ 5, 7, 0 ]} +\section{\textcolor{Chapter }{Random simplicial complexes}}\logpage{[ 5, 10, 0 ]} \hyperdef{L}{X87AF06677F05C624}{} { @@ -3680,7 +3938,7 @@ \section{\textcolor{Chapter }{Random simplicial complexes}}\logpage{[ 5, 7, 0 ]} } -\section{\textcolor{Chapter }{Computing homology of a clique complex (Vietoris-Rips complex) }}\logpage{[ 5, 8, 0 ]} +\section{\textcolor{Chapter }{Computing homology of a clique complex (Vietoris-Rips complex) }}\logpage{[ 5, 11, 0 ]} \hyperdef{L}{X875EE92F7DBA1E27}{} { Topological data analysis provides one motivation for wanting to compute the @@ -3856,7 +4114,85 @@ \section{\textcolor{Chapter }{Aspherical $2$-complexes}}\logpage{[ 6, 4, 0 ]} } -\section{\textcolor{Chapter }{Bogomolov multiplier}}\logpage{[ 6, 5, 0 ]} +\section{\textcolor{Chapter }{Group presentations and homotopical syzygies}}\logpage{[ 6, 5, 0 ]} +\hyperdef{L}{X84C0CB8B7C21E179}{} +{ + Free resolutons for a group $G$ are constructed in \textsc{HAP} as the cellular chain complex $R_\ast=C_\ast(\tilde X)$ of the universal cover of some CW-complex $X=K(G,1)$. The $2$-skeleton of $X$ gives rise to a free presentation for the group $G$. This presentation depends on a choice of maximal tree in the $1$-skeleton of $X$ in cases where $X$ has more than one $0$-cell. The attaching maps of $3$-cells in $X$ can be regarded as \emph{homotopical syzygies} or van Kampen diagrams over the group presentation whose boundaries spell the +trivial word. + +The following example constructs four terms of a resolution for the free +abelian group $G$ on $n=3$ generators, and then extracts the group presentation from the resolution as +well as the unique homotopical syzygy. The syzygy is visualized in terms of +its graph of edges, directed edges being coloured according to the +corresponding group generator. (In this example the CW-complex $\tilde X$ is regular, but in cases where it is not the visualization may be a quotient +of the $1$-skeleton of the syzygy.) +\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] + !gapprompt@gap>| !gapinput@n:=3;;c:=1;;| + !gapprompt@gap>| !gapinput@G:=Image(NqEpimorphismNilpotentQuotient(FreeGroup(n),c));;| + !gapprompt@gap>| !gapinput@R:=ResolutionNilpotentGroup(G,4);;| + !gapprompt@gap>| !gapinput@P:=PresentationOfResolution(R);;| + !gapprompt@gap>| !gapinput@P.freeGroup;| + + !gapprompt@gap>| !gapinput@P.relators;| + [ y^-1*x^-1*y*x, z^-1*x^-1*z*x, z^-1*y^-1*z*y ] + !gapprompt@gap>| !gapinput@IdentityAmongRelatorsDisplay(R,1);| + +\end{Verbatim} + + + + + This homotopical syzygy represents a relationship between the three relators $[x,y]$, $[x,z]$ and $[y,z]$ where $[x,y]=xyx^{-1}y^{-1}$. The syzygy can be thought of as a geometric relationship between commutators +corresponding to the well-known Hall-Witt identity: + +$ [\ [x,y],\ {^yz}\ ]\ \ [\ [y,z],\ {^zx}\ ]\ \ [\ [z,x],\ {^xy}\ ]\ \ =\ \ 1\ \ +.$ + +The homotopical syzygy is special since in this example the edge directions +and labels can be understood as specifying three homeomorphisms between pairs +of faces. Viewing the syzygy as the boundary of the $3$-ball, by using the homeomorphisms to identify the faces in each face pair we +obtain a quotient CW-complex $M$ involving one vertex, three edges, three $2$-cells and one $3$-cell. The cell structure on the quotient exists because, under the +restrictions of homomorphisms to the edges, any cycle of edges retricts to the +identity map on any given edge. The following result tells us that $M$ is in fact a closed oriented compact $3$-manifold. + +\textsc{Theorem.} [Seifert u. Threlfall, Topologie, p.208] \emph{Let $S^2$ denote the boundary of the $3$-ball $B^3$ and suppose that the sphere $S^2$ is given a regular CW-structure in which the faces are partitioned into a +collection of face pairs. Suppose that for each face pair there is an +orientation reversing homeomorphism between the two faces that sends edges to +edges and vertices to vertices. Suppose that by using these homeomorphisms to +identity face pairs we obtain a (not necessarily regular) CW-structure on the +quotient $M$. Then $M$ is a closed compact orientable manifold if and only if its Euler +characteristic is $\chi(M)=0$.} + +The next commands construct a presentation and associated unique homotopical +syzygy for the free nilpotent group of class $c=2$ on $n=2$ generators. +\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] + !gapprompt@gap>| !gapinput@n:=2;;c:=2;;| + !gapprompt@gap>| !gapinput@G:=Image(NqEpimorphismNilpotentQuotient(FreeGroup(n),c));;| + !gapprompt@gap>| !gapinput@R:=ResolutionNilpotentGroup(G,4);;| + !gapprompt@gap>| !gapinput@P:=PresentationOfResolution(R);;| + !gapprompt@gap>| !gapinput@P.freeGroup;| + + !gapprompt@gap>| !gapinput@P.relators;| + [ z*x*y*x^-1*y^-1, z*x*z^-1*x^-1, z*y*z^-1*y^-1 ] + !gapprompt@gap>| !gapinput@IdentityAmongRelatorsDisplay(R,1);| + +\end{Verbatim} + + + + +The syzygy represents the following relationship between commutators (in a +free group). + +$ [\ [x^{-1},y][x,y]\ ,\ [y,x][y^{-1},x]y^{-1}\ ]\ [\ [y,x][y^{-1},x]\ , \ +x^{-1} \ ] \ \ =\ \ 1$ + + Again, using the theorem of Seifert and Threlfall we see that the free +nilpotent group of class two on two generators arises as the fundamental group +of a closed compact orientable $3$-manifold $M$. } + + +\section{\textcolor{Chapter }{Bogomolov multiplier}}\logpage{[ 6, 6, 0 ]} \hyperdef{L}{X7F719758856A443D}{} { @@ -3879,7 +4215,7 @@ \section{\textcolor{Chapter }{Bogomolov multiplier}}\logpage{[ 6, 5, 0 ]} \section{\textcolor{Chapter }{Second group cohomology and group extensions}}\label{secExtensions} -\logpage{[ 6, 6, 0 ]} +\logpage{[ 6, 7, 0 ]} \hyperdef{L}{X8333413B838D787D}{} { Any group extension $N\rightarrowtail E \twoheadrightarrow G$ gives rise to: @@ -4012,7 +4348,7 @@ \section{\textcolor{Chapter }{Second group cohomology and group extensions}}\lab \section{\textcolor{Chapter }{Second group cohomology and cocyclic Hadamard matrices}}\label{secHadamard} -\logpage{[ 6, 7, 0 ]} +\logpage{[ 6, 8, 0 ]} \hyperdef{L}{X7C60E2B578074532}{} { An \emph{Hadamard matrix} is a square $n\times n$ matrix $H$ whose entries are either $+1$ or $-1$ and whose rows are mutually orthogonal, that is $H H^t = nI_n$ where $H^t$ denotes the transpose and $I_n$ denotes the $n\times n$ identity matrix. @@ -4031,7 +4367,7 @@ \section{\textcolor{Chapter }{Second group cohomology and cocyclic Hadamard matr \section{\textcolor{Chapter }{Third group cohomology and homotopy $2$-types}}\label{secCat1} -\logpage{[ 6, 8, 0 ]} +\logpage{[ 6, 9, 0 ]} \hyperdef{L}{X78040D8580D35D53}{} { \textsc{Homotopy 2-types} @@ -4105,8 +4441,8 @@ \section{\textcolor{Chapter }{Third group cohomology and homotopy $2$-types}}\la set-theoretic 'section' of the crossed module corresponding to $H$. } -\chapter{\textcolor{Chapter }{Cohomology of groups}}\logpage{[ 7, 0, 0 ]} -\hyperdef{L}{X7E34E2C6868F2726}{} +\chapter{\textcolor{Chapter }{Cohomology of groups (and Lie Algebras)}}\logpage{[ 7, 0, 0 ]} +\hyperdef{L}{X787E37187B7308C9}{} { \section{\textcolor{Chapter }{Finite groups }}\logpage{[ 7, 1, 0 ]} @@ -5118,6 +5454,27 @@ \section{\textcolor{Chapter }{Quillen's complex and the $p$-part of homology }}\ \end{Verbatim} } + +\section{\textcolor{Chapter }{Homology of a Lie algebra with coefficients in a module}}\logpage{[ 7, 15, 0 ]} +\hyperdef{L}{X83F9A1A184FB3475}{} +{ + Let $A$ be the Lie algebra constructed from the associative algebra $M^{4\times 4}(\mathbb Q)$ of all $4\times 4$ rational matrices. Let $V$ be its adjoint module (with underlying vector space of dimension $16$ and equal to that of $A$). The following commands compute $H_{4}(A,V) = \mathbb Q$. +\begin{Verbatim}[commandchars=@|B,fontsize=\small,frame=single,label=Example] + @gapprompt|gap>B @gapinput|M:=FullMatrixAlgebra(Rationals,4);; B + @gapprompt|gap>B @gapinput|A:=LieAlgebra(M);;B + @gapprompt|gap>B @gapinput|V:=AdjointModule(A);;B + @gapprompt|gap>B @gapinput|C:=ChevalleyEilenbergComplex(V,17);;B + @gapprompt|gap>B @gapinput|List([0..17],C!.dimension);B + [ 16, 256, 1920, 8960, 29120, 69888, 128128, 183040, 205920, 183040, 128128, + 69888, 29120, 8960, 1920, 256, 16, 0 ] + @gapprompt|gap>B @gapinput|Homology(C,4);B + 1 + +\end{Verbatim} + + +Note that the eighth term $C_{8}(V)$ in the Chevalley-Eilenberg complex $C_\ast(V)$ is a vector space of dimension $205920$ and so it will take longer to compute the homology in degree $8$. } + } @@ -5905,10 +6262,11 @@ \section{\textcolor{Chapter }{Resolutions for very small finite groups}}\logpage [ 4, 42, 8, 128, 4, 42, 8, 128, 4, 42, 8, 128, 4, 42, 8, 128, 4, 42, 8, 128 ] \end{Verbatim} - The suspicion that this resolution $R_\ast$ is periodic of period $4$ can be verified by constructing the chain complex $C_\ast=R_\ast\otimes_{\mathbb Z}\mathbb ZG$ and verifying that boundary matrices repeat with period $4$. + The suspicion that this resolution $R_\ast$ is periodic of period $4$ can be confirmed by constructing the chain complex $C_\ast=R_\ast\otimes_{\mathbb Z}\mathbb ZG$ and verifying that boundary matrices repeat with period $4$. - A second example of a periodic resolution, for the Dihedral group $D_{2k+1}=\langle x, y\ |\ x^2= xy^kx^{-1}y^{-k-1}\rangle$ of order $2k+2$ in the case $k=1$, is constructed and verified for periodicity in the next example. + A second example of a periodic resolution, for the Dihedral group $D_{2k+1}=\langle x, y\ |\ x^2= xy^kx^{-1}y^{-k-1} = 1\rangle$ of order $2k+2$ in the case $k=1$, is constructed and verified for periodicity in the next example. \begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] + !gapprompt@gap>| !gapinput@F:=FreeGroup(2);;D:=F/[F.1^2,F.1*F.2*F.1^-1*F.2^-2];;| !gapprompt@gap>| !gapinput@R:=ResolutionSmallGroup(D,15);;| !gapprompt@gap>| !gapinput@Size(R);| [ 4, 7, 8, 6, 4, 8, 8, 6, 4, 8, 8, 6, 4, 8, 8 ] @@ -5939,6 +6297,29 @@ \section{\textcolor{Chapter }{Resolutions for very small finite groups}}\logpage @gapprompt|gap>A @gapinput|List([0..7],R!.dimension);A [ 1, 2, 2, 2, 2, 2, 2, 2 ] +\end{Verbatim} + + +The performance of the function \texttt{ResolutionSmallGroup(G,n)} is very sensistive to the choice of presentation for the input group $G$. If $G$ is an fp-group then the defining presentation for $G$ is used. If $G$ is a permutaion group or finite matrix group then \textsc{GAP} functions are invoked to find a presentation for $G$. The following commands use a geometrically derived presentation for $SL(2,5)$ as input in order to obtain the first few terms of a periodic resolution for +this group of period $4$. +\begin{Verbatim}[commandchars=@|A,fontsize=\small,frame=single,label=Example] + @gapprompt|gap>A @gapinput|Y:=PoincareDodecahedronCWComplex( A + @gapprompt|>A @gapinput|[[1,2,3,4,5],[6,7,8,9,10]],A + @gapprompt|>A @gapinput|[[1,11,16,12,2],[19,9,8,18,14]],A + @gapprompt|>A @gapinput|[[2,12,17,13,3],[20,10,9,19,15]],A + @gapprompt|>A @gapinput|[[3,13,18,14,4],[16,6,10,20,11]],A + @gapprompt|>A @gapinput|[[4,14,19,15,5],[17,7,6,16,12]],A + @gapprompt|>A @gapinput|[[5,15,20,11,1],[18,8,7,17,13]]);;A + @gapprompt|gap>A @gapinput|G:=FundamentalGroup(Y);A + + @gapprompt|gap>A @gapinput|RelatorsOfFpGroup(G);A + [ f2^-1*f1^-1*f2*f1^-1*f2^-1*f1, f2^-1*f1*f2^2*f1*f2^-1*f1^-1 ] + @gapprompt|gap>A @gapinput|StructureDescription(G);A + "SL(2,5)" + @gapprompt|gap>A @gapinput|R:=ResolutionSmallGroup(G,3);;A + @gapprompt|gap>A @gapinput|List([0..3],R!.dimension); A + [ 1, 2, 2, 1 ] + \end{Verbatim} } diff --git a/tutorial/HapTutorial.toc b/tutorial/HapTutorial.toc index b82f133a..88e294b8 100644 --- a/tutorial/HapTutorial.toc +++ b/tutorial/HapTutorial.toc @@ -1,176 +1,183 @@ -\contentsline {chapter}{\numberline {1}\textcolor {Chapter }{Simplicial complexes \& CW complexes}}{6}{chapter.1}% -\contentsline {section}{\numberline {1.1}\textcolor {Chapter }{The Klein bottle as a simplicial complex}}{6}{section.1.1}% -\contentsline {section}{\numberline {1.2}\textcolor {Chapter }{Other simplicial surfaces}}{7}{section.1.2}% -\contentsline {section}{\numberline {1.3}\textcolor {Chapter }{The Quillen complex}}{7}{section.1.3}% -\contentsline {section}{\numberline {1.4}\textcolor {Chapter }{The Quillen complex as a reduced CW-complex}}{8}{section.1.4}% -\contentsline {section}{\numberline {1.5}\textcolor {Chapter }{Simple homotopy equivalences}}{8}{section.1.5}% -\contentsline {section}{\numberline {1.6}\textcolor {Chapter }{Cellular simplifications preserving homeomorphism type}}{9}{section.1.6}% -\contentsline {section}{\numberline {1.7}\textcolor {Chapter }{Constructing a CW-structure on a knot complement}}{9}{section.1.7}% -\contentsline {section}{\numberline {1.8}\textcolor {Chapter }{Constructing a regular CW-complex by attaching cells}}{10}{section.1.8}% -\contentsline {section}{\numberline {1.9}\textcolor {Chapter }{Constructing a regular CW-complex from its face lattice}}{11}{section.1.9}% -\contentsline {section}{\numberline {1.10}\textcolor {Chapter }{Cup products}}{12}{section.1.10}% -\contentsline {section}{\numberline {1.11}\textcolor {Chapter }{Intersection forms of $4$-manifolds}}{17}{section.1.11}% -\contentsline {section}{\numberline {1.12}\textcolor {Chapter }{Cohomology Rings}}{18}{section.1.12}% -\contentsline {section}{\numberline {1.13}\textcolor {Chapter }{Bockstein homomorphism}}{19}{section.1.13}% -\contentsline {section}{\numberline {1.14}\textcolor {Chapter }{CW maps and induced homomorphisms}}{20}{section.1.14}% -\contentsline {section}{\numberline {1.15}\textcolor {Chapter }{Constructing a simplicial complex from a regular CW-complex}}{20}{section.1.15}% -\contentsline {section}{\numberline {1.16}\textcolor {Chapter }{Equivariant CW complexes}}{21}{section.1.16}% -\contentsline {section}{\numberline {1.17}\textcolor {Chapter }{Orbifolds and classifying spaces}}{23}{section.1.17}% -\contentsline {chapter}{\numberline {2}\textcolor {Chapter }{Cubical complexes \& permutahedral complexes}}{27}{chapter.2}% -\contentsline {section}{\numberline {2.1}\textcolor {Chapter }{Cubical complexes}}{27}{section.2.1}% -\contentsline {section}{\numberline {2.2}\textcolor {Chapter }{Permutahedral complexes}}{28}{section.2.2}% -\contentsline {section}{\numberline {2.3}\textcolor {Chapter }{Constructing pure cubical and permutahedral complexes}}{30}{section.2.3}% -\contentsline {section}{\numberline {2.4}\textcolor {Chapter }{Computations in dynamical systems}}{31}{section.2.4}% -\contentsline {chapter}{\numberline {3}\textcolor {Chapter }{Covering spaces}}{32}{chapter.3}% -\contentsline {section}{\numberline {3.1}\textcolor {Chapter }{Cellular chains on the universal cover}}{32}{section.3.1}% -\contentsline {section}{\numberline {3.2}\textcolor {Chapter }{Spun knots and the Satoh tube map}}{33}{section.3.2}% -\contentsline {section}{\numberline {3.3}\textcolor {Chapter }{Cohomology with local coefficients}}{35}{section.3.3}% -\contentsline {section}{\numberline {3.4}\textcolor {Chapter }{Distinguishing between two non-homeomorphic homotopy equivalent spaces}}{36}{section.3.4}% -\contentsline {section}{\numberline {3.5}\textcolor {Chapter }{ Second homotopy groups of spaces with finite fundamental group}}{36}{section.3.5}% -\contentsline {section}{\numberline {3.6}\textcolor {Chapter }{Third homotopy groups of simply connected spaces}}{37}{section.3.6}% -\contentsline {subsection}{\numberline {3.6.1}\textcolor {Chapter }{First example: Whitehead's certain exact sequence}}{37}{subsection.3.6.1}% -\contentsline {subsection}{\numberline {3.6.2}\textcolor {Chapter }{Second example: the Hopf invariant}}{38}{subsection.3.6.2}% -\contentsline {section}{\numberline {3.7}\textcolor {Chapter }{Computing the second homotopy group of a space with infinite fundamental group}}{39}{section.3.7}% -\contentsline {chapter}{\numberline {4}\textcolor {Chapter }{Three Manifolds}}{41}{chapter.4}% -\contentsline {section}{\numberline {4.1}\textcolor {Chapter }{Dehn Surgery}}{41}{section.4.1}% -\contentsline {section}{\numberline {4.2}\textcolor {Chapter }{Connected Sums}}{42}{section.4.2}% -\contentsline {section}{\numberline {4.3}\textcolor {Chapter }{Dijkgraaf-Witten Invariant}}{42}{section.4.3}% -\contentsline {section}{\numberline {4.4}\textcolor {Chapter }{Cohomology rings}}{43}{section.4.4}% -\contentsline {section}{\numberline {4.5}\textcolor {Chapter }{Linking Form}}{44}{section.4.5}% -\contentsline {section}{\numberline {4.6}\textcolor {Chapter }{Determining the homeomorphism type of a lens space}}{45}{section.4.6}% -\contentsline {section}{\numberline {4.7}\textcolor {Chapter }{Surgeries on distinct knots can yield homeomorphic manifolds}}{47}{section.4.7}% -\contentsline {section}{\numberline {4.8}\textcolor {Chapter }{Finite fundamental groups of $3$-manifolds}}{48}{section.4.8}% -\contentsline {section}{\numberline {4.9}\textcolor {Chapter }{Poincare's cube manifolds}}{49}{section.4.9}% -\contentsline {section}{\numberline {4.10}\textcolor {Chapter }{There are at least 25 distinct cube manifolds}}{50}{section.4.10}% -\contentsline {subsection}{\numberline {4.10.1}\textcolor {Chapter }{Face pairings for 25 distinct cube manifolds}}{52}{subsection.4.10.1}% -\contentsline {subsection}{\numberline {4.10.2}\textcolor {Chapter }{Platonic cube manifolds}}{56}{subsection.4.10.2}% -\contentsline {section}{\numberline {4.11}\textcolor {Chapter }{There are at most 41 distinct cube manifolds}}{56}{section.4.11}% -\contentsline {section}{\numberline {4.12}\textcolor {Chapter }{There are precisely 18 orientable cube manifolds, of which 9 are spherical and 5 are euclidean}}{58}{section.4.12}% -\contentsline {section}{\numberline {4.13}\textcolor {Chapter }{Cube manifolds with boundary}}{60}{section.4.13}% -\contentsline {section}{\numberline {4.14}\textcolor {Chapter }{Octahedral manifolds}}{61}{section.4.14}% -\contentsline {section}{\numberline {4.15}\textcolor {Chapter }{Dodecahedral manifolds}}{61}{section.4.15}% -\contentsline {section}{\numberline {4.16}\textcolor {Chapter }{Prism manifolds}}{62}{section.4.16}% -\contentsline {section}{\numberline {4.17}\textcolor {Chapter }{Bipyramid manifolds}}{63}{section.4.17}% -\contentsline {chapter}{\numberline {5}\textcolor {Chapter }{Topological data analysis}}{64}{chapter.5}% -\contentsline {section}{\numberline {5.1}\textcolor {Chapter }{Persistent homology }}{64}{section.5.1}% -\contentsline {subsection}{\numberline {5.1.1}\textcolor {Chapter }{Background to the data}}{65}{subsection.5.1.1}% -\contentsline {section}{\numberline {5.2}\textcolor {Chapter }{Mapper clustering}}{65}{section.5.2}% -\contentsline {subsection}{\numberline {5.2.1}\textcolor {Chapter }{Background to the data}}{66}{subsection.5.2.1}% -\contentsline {section}{\numberline {5.3}\textcolor {Chapter }{Digital image analysis and persistent homology}}{66}{section.5.3}% -\contentsline {subsection}{\numberline {5.3.1}\textcolor {Chapter }{Naive example of image segmentation by automatic thresholding}}{66}{subsection.5.3.1}% -\contentsline {subsection}{\numberline {5.3.2}\textcolor {Chapter }{Background to the data}}{66}{subsection.5.3.2}% -\contentsline {section}{\numberline {5.4}\textcolor {Chapter }{A second example of digital image segmentation}}{67}{section.5.4}% -\contentsline {section}{\numberline {5.5}\textcolor {Chapter }{Alternative approaches to computing persistent homology}}{67}{section.5.5}% -\contentsline {subsection}{\numberline {5.5.1}\textcolor {Chapter }{Non-trivial cup product}}{68}{subsection.5.5.1}% -\contentsline {section}{\numberline {5.6}\textcolor {Chapter }{Knotted proteins}}{69}{section.5.6}% -\contentsline {section}{\numberline {5.7}\textcolor {Chapter }{Random simplicial complexes}}{70}{section.5.7}% -\contentsline {section}{\numberline {5.8}\textcolor {Chapter }{Computing homology of a clique complex (Vietoris-Rips complex) }}{72}{section.5.8}% -\contentsline {chapter}{\numberline {6}\textcolor {Chapter }{Group theoretic computations}}{74}{chapter.6}% -\contentsline {section}{\numberline {6.1}\textcolor {Chapter }{Third homotopy group of a supsension of an Eilenberg-MacLane space }}{74}{section.6.1}% -\contentsline {section}{\numberline {6.2}\textcolor {Chapter }{Representations of knot quandles}}{74}{section.6.2}% -\contentsline {section}{\numberline {6.3}\textcolor {Chapter }{Identifying knots}}{75}{section.6.3}% -\contentsline {section}{\numberline {6.4}\textcolor {Chapter }{Aspherical $2$-complexes}}{75}{section.6.4}% -\contentsline {section}{\numberline {6.5}\textcolor {Chapter }{Bogomolov multiplier}}{76}{section.6.5}% -\contentsline {section}{\numberline {6.6}\textcolor {Chapter }{Second group cohomology and group extensions}}{76}{section.6.6}% -\contentsline {section}{\numberline {6.7}\textcolor {Chapter }{Second group cohomology and cocyclic Hadamard matrices}}{78}{section.6.7}% -\contentsline {section}{\numberline {6.8}\textcolor {Chapter }{Third group cohomology and homotopy $2$-types}}{79}{section.6.8}% -\contentsline {chapter}{\numberline {7}\textcolor {Chapter }{Cohomology of groups}}{81}{chapter.7}% -\contentsline {section}{\numberline {7.1}\textcolor {Chapter }{Finite groups }}{81}{section.7.1}% -\contentsline {subsection}{\numberline {7.1.1}\textcolor {Chapter }{Naive homology computation for a very small group}}{81}{subsection.7.1.1}% -\contentsline {subsection}{\numberline {7.1.2}\textcolor {Chapter }{A more efficient homology computation}}{81}{subsection.7.1.2}% -\contentsline {subsection}{\numberline {7.1.3}\textcolor {Chapter }{Computation of an induced homology homomorphism}}{82}{subsection.7.1.3}% -\contentsline {subsection}{\numberline {7.1.4}\textcolor {Chapter }{Some other finite group homology computations}}{82}{subsection.7.1.4}% -\contentsline {section}{\numberline {7.2}\textcolor {Chapter }{Nilpotent groups}}{84}{section.7.2}% -\contentsline {section}{\numberline {7.3}\textcolor {Chapter }{Crystallographic and Almost Crystallographic groups}}{84}{section.7.3}% -\contentsline {section}{\numberline {7.4}\textcolor {Chapter }{Arithmetic groups}}{85}{section.7.4}% -\contentsline {section}{\numberline {7.5}\textcolor {Chapter }{Artin groups}}{85}{section.7.5}% -\contentsline {section}{\numberline {7.6}\textcolor {Chapter }{Graphs of groups}}{85}{section.7.6}% -\contentsline {section}{\numberline {7.7}\textcolor {Chapter }{Lie algebra homology and free nilpotent groups}}{86}{section.7.7}% -\contentsline {section}{\numberline {7.8}\textcolor {Chapter }{Cohomology with coefficients in a module}}{87}{section.7.8}% -\contentsline {section}{\numberline {7.9}\textcolor {Chapter }{Cohomology as a functor of the first variable}}{89}{section.7.9}% -\contentsline {section}{\numberline {7.10}\textcolor {Chapter }{Cohomology as a functor of the second variable and the long exact coefficient sequence}}{90}{section.7.10}% -\contentsline {section}{\numberline {7.11}\textcolor {Chapter }{Transfer Homomorphism}}{91}{section.7.11}% -\contentsline {section}{\numberline {7.12}\textcolor {Chapter }{Cohomology rings of finite fundamental groups of 3-manifolds }}{91}{section.7.12}% -\contentsline {section}{\numberline {7.13}\textcolor {Chapter }{Explicit cocycles }}{93}{section.7.13}% -\contentsline {section}{\numberline {7.14}\textcolor {Chapter }{Quillen's complex and the $p$-part of homology }}{96}{section.7.14}% -\contentsline {chapter}{\numberline {8}\textcolor {Chapter }{Cohomology rings and Steenrod operations for finite groups}}{99}{chapter.8}% -\contentsline {section}{\numberline {8.1}\textcolor {Chapter }{Mod-$p$ cohomology rings of finite groups}}{99}{section.8.1}% -\contentsline {subsection}{\numberline {8.1.1}\textcolor {Chapter }{Ring presentations (for the commutative $p=2$ case)}}{100}{subsection.8.1.1}% -\contentsline {section}{\numberline {8.2}\textcolor {Chapter }{Functorial ring homomorphisms in Mod-$p$ cohomology}}{100}{section.8.2}% -\contentsline {subsection}{\numberline {8.2.1}\textcolor {Chapter }{Testing homomorphism properties}}{100}{subsection.8.2.1}% -\contentsline {subsection}{\numberline {8.2.2}\textcolor {Chapter }{Testing functorial properties}}{101}{subsection.8.2.2}% -\contentsline {subsection}{\numberline {8.2.3}\textcolor {Chapter }{Computing with larger groups}}{101}{subsection.8.2.3}% -\contentsline {section}{\numberline {8.3}\textcolor {Chapter }{Cohomology rings of finite $2$-groups}}{102}{section.8.3}% -\contentsline {section}{\numberline {8.4}\textcolor {Chapter }{Steenrod operations for finite $2$-groups}}{103}{section.8.4}% -\contentsline {section}{\numberline {8.5}\textcolor {Chapter }{Steenrod operations on the classifying space of a finite $p$-group}}{104}{section.8.5}% -\contentsline {chapter}{\numberline {9}\textcolor {Chapter }{Bredon homology}}{105}{chapter.9}% -\contentsline {section}{\numberline {9.1}\textcolor {Chapter }{Davis complex}}{105}{section.9.1}% -\contentsline {section}{\numberline {9.2}\textcolor {Chapter }{Arithmetic groups}}{105}{section.9.2}% -\contentsline {section}{\numberline {9.3}\textcolor {Chapter }{Crystallographic groups}}{106}{section.9.3}% -\contentsline {chapter}{\numberline {10}\textcolor {Chapter }{Chain Complexes}}{107}{chapter.10}% -\contentsline {section}{\numberline {10.1}\textcolor {Chapter }{Chain complex of a simplicial complex and simplicial pair}}{107}{section.10.1}% -\contentsline {section}{\numberline {10.2}\textcolor {Chapter }{Chain complex of a cubical complex and cubical pair}}{108}{section.10.2}% -\contentsline {section}{\numberline {10.3}\textcolor {Chapter }{Chain complex of a regular CW-complex}}{109}{section.10.3}% -\contentsline {section}{\numberline {10.4}\textcolor {Chapter }{Chain Maps of simplicial and regular CW maps}}{110}{section.10.4}% -\contentsline {section}{\numberline {10.5}\textcolor {Chapter }{Constructions for chain complexes}}{110}{section.10.5}% -\contentsline {section}{\numberline {10.6}\textcolor {Chapter }{Filtered chain complexes}}{111}{section.10.6}% -\contentsline {section}{\numberline {10.7}\textcolor {Chapter }{Sparse chain complexes}}{112}{section.10.7}% -\contentsline {chapter}{\numberline {11}\textcolor {Chapter }{Resolutions}}{114}{chapter.11}% -\contentsline {section}{\numberline {11.1}\textcolor {Chapter }{Resolutions for small finite groups}}{114}{section.11.1}% -\contentsline {section}{\numberline {11.2}\textcolor {Chapter }{Resolutions for very small finite groups}}{114}{section.11.2}% -\contentsline {section}{\numberline {11.3}\textcolor {Chapter }{Resolutions for finite groups acting on orbit polytopes}}{115}{section.11.3}% -\contentsline {section}{\numberline {11.4}\textcolor {Chapter }{Minimal resolutions for finite $p$-groups over $\mathbb F_p$}}{116}{section.11.4}% -\contentsline {section}{\numberline {11.5}\textcolor {Chapter }{Resolutions for abelian groups}}{117}{section.11.5}% -\contentsline {section}{\numberline {11.6}\textcolor {Chapter }{Resolutions for nilpotent groups}}{117}{section.11.6}% -\contentsline {section}{\numberline {11.7}\textcolor {Chapter }{Resolutions for groups with subnormal series}}{118}{section.11.7}% -\contentsline {section}{\numberline {11.8}\textcolor {Chapter }{Resolutions for groups with normal series}}{118}{section.11.8}% -\contentsline {section}{\numberline {11.9}\textcolor {Chapter }{Resolutions for polycyclic (almost) crystallographic groups }}{119}{section.11.9}% -\contentsline {section}{\numberline {11.10}\textcolor {Chapter }{Resolutions for Bieberbach groups }}{120}{section.11.10}% -\contentsline {section}{\numberline {11.11}\textcolor {Chapter }{Resolutions for arbitrary crystallographic groups}}{121}{section.11.11}% -\contentsline {section}{\numberline {11.12}\textcolor {Chapter }{Resolutions for crystallographic groups admitting cubical fundamental domain}}{121}{section.11.12}% -\contentsline {section}{\numberline {11.13}\textcolor {Chapter }{Resolutions for Coxeter groups }}{122}{section.11.13}% -\contentsline {section}{\numberline {11.14}\textcolor {Chapter }{Resolutions for Artin groups }}{122}{section.11.14}% -\contentsline {section}{\numberline {11.15}\textcolor {Chapter }{Resolutions for $G=SL_2(\mathbb Z[1/m])$}}{122}{section.11.15}% -\contentsline {section}{\numberline {11.16}\textcolor {Chapter }{Resolutions for selected groups $G=SL_2( {\mathcal O}(\mathbb Q(\sqrt {d}) )$}}{123}{section.11.16}% -\contentsline {section}{\numberline {11.17}\textcolor {Chapter }{Resolutions for selected groups $G=PSL_2( {\mathcal O}(\mathbb Q(\sqrt {d}) )$}}{123}{section.11.17}% -\contentsline {section}{\numberline {11.18}\textcolor {Chapter }{Resolutions for a few higher-dimensional arithmetic groups }}{123}{section.11.18}% -\contentsline {section}{\numberline {11.19}\textcolor {Chapter }{Resolutions for finite-index subgroups }}{124}{section.11.19}% -\contentsline {section}{\numberline {11.20}\textcolor {Chapter }{Simplifying resolutions }}{124}{section.11.20}% -\contentsline {section}{\numberline {11.21}\textcolor {Chapter }{Resolutions for graphs of groups and for groups with aspherical presentations }}{125}{section.11.21}% -\contentsline {section}{\numberline {11.22}\textcolor {Chapter }{Resolutions for $\mathbb FG$-modules }}{125}{section.11.22}% -\contentsline {chapter}{\numberline {12}\textcolor {Chapter }{Simplicial groups}}{127}{chapter.12}% -\contentsline {section}{\numberline {12.1}\textcolor {Chapter }{Crossed modules}}{127}{section.12.1}% -\contentsline {section}{\numberline {12.2}\textcolor {Chapter }{Eilenberg-MacLane spaces as simplicial groups (not recommended)}}{128}{section.12.2}% -\contentsline {section}{\numberline {12.3}\textcolor {Chapter }{Eilenberg-MacLane spaces as simplicial free abelian groups (recommended)}}{128}{section.12.3}% -\contentsline {section}{\numberline {12.4}\textcolor {Chapter }{Elementary theoretical information on $H^\ast (K(\pi ,n),\mathbb Z)$}}{130}{section.12.4}% -\contentsline {section}{\numberline {12.5}\textcolor {Chapter }{The first three non-trivial homotopy groups of spheres}}{131}{section.12.5}% -\contentsline {section}{\numberline {12.6}\textcolor {Chapter }{The first two non-trivial homotopy groups of the suspension and double suspension of a $K(G,1)$}}{132}{section.12.6}% -\contentsline {section}{\numberline {12.7}\textcolor {Chapter }{Postnikov towers and $\pi _5(S^3)$ }}{132}{section.12.7}% -\contentsline {section}{\numberline {12.8}\textcolor {Chapter }{Towards $\pi _4(\Sigma K(G,1))$ }}{134}{section.12.8}% -\contentsline {section}{\numberline {12.9}\textcolor {Chapter }{Enumerating homotopy 2-types}}{135}{section.12.9}% -\contentsline {section}{\numberline {12.10}\textcolor {Chapter }{Identifying cat$^1$-groups of low order}}{136}{section.12.10}% -\contentsline {section}{\numberline {12.11}\textcolor {Chapter }{Identifying crossed modules of low order}}{137}{section.12.11}% -\contentsline {chapter}{\numberline {13}\textcolor {Chapter }{Congruence Subgroups, Cuspidal Cohomology and Hecke Operators}}{139}{chapter.13}% -\contentsline {section}{\numberline {13.1}\textcolor {Chapter }{Eichler-Shimura isomorphism}}{139}{section.13.1}% -\contentsline {section}{\numberline {13.2}\textcolor {Chapter }{Generators for $SL_2(\mathbb Z)$ and the cubic tree}}{140}{section.13.2}% -\contentsline {section}{\numberline {13.3}\textcolor {Chapter }{One-dimensional fundamental domains and generators for congruence subgroups}}{141}{section.13.3}% -\contentsline {section}{\numberline {13.4}\textcolor {Chapter }{Cohomology of congruence subgroups}}{142}{section.13.4}% -\contentsline {subsection}{\numberline {13.4.1}\textcolor {Chapter }{Cohomology with rational coefficients}}{144}{subsection.13.4.1}% -\contentsline {section}{\numberline {13.5}\textcolor {Chapter }{Cuspidal cohomology}}{144}{section.13.5}% -\contentsline {section}{\numberline {13.6}\textcolor {Chapter }{Hecke operators on forms of weight 2}}{146}{section.13.6}% -\contentsline {section}{\numberline {13.7}\textcolor {Chapter }{Hecke operators on forms of weight $ \ge 2$}}{147}{section.13.7}% -\contentsline {section}{\numberline {13.8}\textcolor {Chapter }{Reconstructing modular forms from cohomology computations}}{147}{section.13.8}% -\contentsline {section}{\numberline {13.9}\textcolor {Chapter }{The Picard group}}{149}{section.13.9}% -\contentsline {section}{\numberline {13.10}\textcolor {Chapter }{Bianchi groups}}{151}{section.13.10}% -\contentsline {section}{\numberline {13.11}\textcolor {Chapter }{Some other infinite matrix groups}}{152}{section.13.11}% -\contentsline {section}{\numberline {13.12}\textcolor {Chapter }{Ideals and finite quotient groups}}{154}{section.13.12}% -\contentsline {section}{\numberline {13.13}\textcolor {Chapter }{Congruence subgroups for ideals}}{156}{section.13.13}% -\contentsline {section}{\numberline {13.14}\textcolor {Chapter }{First homology}}{157}{section.13.14}% -\contentsline {chapter}{\numberline {14}\textcolor {Chapter }{Parallel computation}}{159}{chapter.14}% -\contentsline {section}{\numberline {14.1}\textcolor {Chapter }{An embarassingly parallel computation}}{159}{section.14.1}% -\contentsline {section}{\numberline {14.2}\textcolor {Chapter }{An non-embarassingly parallel computation}}{159}{section.14.2}% -\contentsline {chapter}{\numberline {15}\textcolor {Chapter }{Regular CW-structure on knots (written by Kelvin Killeen)}}{161}{chapter.15}% -\contentsline {section}{\numberline {15.1}\textcolor {Chapter }{Knot complements in the 3-ball}}{161}{section.15.1}% -\contentsline {section}{\numberline {15.2}\textcolor {Chapter }{Tubular neighbourhoods}}{162}{section.15.2}% -\contentsline {section}{\numberline {15.3}\textcolor {Chapter }{Knotted surface complements in the 4-ball}}{165}{section.15.3}% -\contentsline {chapter}{References}{174}{chapter*.2}% +\contentsline {chapter}{\numberline {1}\textcolor {Chapter }{Simplicial complexes \& CW complexes}}{7}{chapter.1}% +\contentsline {section}{\numberline {1.1}\textcolor {Chapter }{The Klein bottle as a simplicial complex}}{7}{section.1.1}% +\contentsline {section}{\numberline {1.2}\textcolor {Chapter }{Other simplicial surfaces}}{8}{section.1.2}% +\contentsline {section}{\numberline {1.3}\textcolor {Chapter }{The Quillen complex}}{8}{section.1.3}% +\contentsline {section}{\numberline {1.4}\textcolor {Chapter }{The Quillen complex as a reduced CW-complex}}{9}{section.1.4}% +\contentsline {section}{\numberline {1.5}\textcolor {Chapter }{Simple homotopy equivalences}}{9}{section.1.5}% +\contentsline {section}{\numberline {1.6}\textcolor {Chapter }{Cellular simplifications preserving homeomorphism type}}{10}{section.1.6}% +\contentsline {section}{\numberline {1.7}\textcolor {Chapter }{Constructing a CW-structure on a knot complement}}{10}{section.1.7}% +\contentsline {section}{\numberline {1.8}\textcolor {Chapter }{Constructing a regular CW-complex by attaching cells}}{11}{section.1.8}% +\contentsline {section}{\numberline {1.9}\textcolor {Chapter }{Constructing a regular CW-complex from its face lattice}}{12}{section.1.9}% +\contentsline {section}{\numberline {1.10}\textcolor {Chapter }{Cup products}}{13}{section.1.10}% +\contentsline {section}{\numberline {1.11}\textcolor {Chapter }{Intersection forms of $4$-manifolds}}{18}{section.1.11}% +\contentsline {section}{\numberline {1.12}\textcolor {Chapter }{Cohomology Rings}}{19}{section.1.12}% +\contentsline {section}{\numberline {1.13}\textcolor {Chapter }{Bockstein homomorphism}}{20}{section.1.13}% +\contentsline {section}{\numberline {1.14}\textcolor {Chapter }{CW maps and induced homomorphisms}}{21}{section.1.14}% +\contentsline {section}{\numberline {1.15}\textcolor {Chapter }{Constructing a simplicial complex from a regular CW-complex}}{21}{section.1.15}% +\contentsline {section}{\numberline {1.16}\textcolor {Chapter }{Equivariant CW complexes}}{22}{section.1.16}% +\contentsline {section}{\numberline {1.17}\textcolor {Chapter }{Orbifolds and classifying spaces}}{24}{section.1.17}% +\contentsline {chapter}{\numberline {2}\textcolor {Chapter }{Cubical complexes \& permutahedral complexes}}{28}{chapter.2}% +\contentsline {section}{\numberline {2.1}\textcolor {Chapter }{Cubical complexes}}{28}{section.2.1}% +\contentsline {section}{\numberline {2.2}\textcolor {Chapter }{Permutahedral complexes}}{29}{section.2.2}% +\contentsline {section}{\numberline {2.3}\textcolor {Chapter }{Constructing pure cubical and permutahedral complexes}}{31}{section.2.3}% +\contentsline {section}{\numberline {2.4}\textcolor {Chapter }{Computations in dynamical systems}}{32}{section.2.4}% +\contentsline {chapter}{\numberline {3}\textcolor {Chapter }{Covering spaces}}{33}{chapter.3}% +\contentsline {section}{\numberline {3.1}\textcolor {Chapter }{Cellular chains on the universal cover}}{33}{section.3.1}% +\contentsline {section}{\numberline {3.2}\textcolor {Chapter }{Spun knots and the Satoh tube map}}{34}{section.3.2}% +\contentsline {section}{\numberline {3.3}\textcolor {Chapter }{Cohomology with local coefficients}}{36}{section.3.3}% +\contentsline {section}{\numberline {3.4}\textcolor {Chapter }{Distinguishing between two non-homeomorphic homotopy equivalent spaces}}{37}{section.3.4}% +\contentsline {section}{\numberline {3.5}\textcolor {Chapter }{ Second homotopy groups of spaces with finite fundamental group}}{37}{section.3.5}% +\contentsline {section}{\numberline {3.6}\textcolor {Chapter }{Third homotopy groups of simply connected spaces}}{38}{section.3.6}% +\contentsline {subsection}{\numberline {3.6.1}\textcolor {Chapter }{First example: Whitehead's certain exact sequence}}{38}{subsection.3.6.1}% +\contentsline {subsection}{\numberline {3.6.2}\textcolor {Chapter }{Second example: the Hopf invariant}}{39}{subsection.3.6.2}% +\contentsline {section}{\numberline {3.7}\textcolor {Chapter }{Computing the second homotopy group of a space with infinite fundamental group}}{40}{section.3.7}% +\contentsline {chapter}{\numberline {4}\textcolor {Chapter }{Three Manifolds}}{42}{chapter.4}% +\contentsline {section}{\numberline {4.1}\textcolor {Chapter }{Dehn Surgery}}{42}{section.4.1}% +\contentsline {section}{\numberline {4.2}\textcolor {Chapter }{Connected Sums}}{43}{section.4.2}% +\contentsline {section}{\numberline {4.3}\textcolor {Chapter }{Dijkgraaf-Witten Invariant}}{43}{section.4.3}% +\contentsline {section}{\numberline {4.4}\textcolor {Chapter }{Cohomology rings}}{44}{section.4.4}% +\contentsline {section}{\numberline {4.5}\textcolor {Chapter }{Linking Form}}{45}{section.4.5}% +\contentsline {section}{\numberline {4.6}\textcolor {Chapter }{Determining the homeomorphism type of a lens space}}{46}{section.4.6}% +\contentsline {section}{\numberline {4.7}\textcolor {Chapter }{Surgeries on distinct knots can yield homeomorphic manifolds}}{48}{section.4.7}% +\contentsline {section}{\numberline {4.8}\textcolor {Chapter }{Finite fundamental groups of $3$-manifolds}}{49}{section.4.8}% +\contentsline {section}{\numberline {4.9}\textcolor {Chapter }{Poincare's cube manifolds}}{50}{section.4.9}% +\contentsline {section}{\numberline {4.10}\textcolor {Chapter }{There are at least 25 distinct cube manifolds}}{51}{section.4.10}% +\contentsline {subsection}{\numberline {4.10.1}\textcolor {Chapter }{Face pairings for 25 distinct cube manifolds}}{53}{subsection.4.10.1}% +\contentsline {subsection}{\numberline {4.10.2}\textcolor {Chapter }{Platonic cube manifolds}}{57}{subsection.4.10.2}% +\contentsline {section}{\numberline {4.11}\textcolor {Chapter }{There are at most 41 distinct cube manifolds}}{57}{section.4.11}% +\contentsline {section}{\numberline {4.12}\textcolor {Chapter }{There are precisely 18 orientable cube manifolds, of which 9 are spherical and 5 are euclidean}}{59}{section.4.12}% +\contentsline {section}{\numberline {4.13}\textcolor {Chapter }{Cube manifolds with boundary}}{61}{section.4.13}% +\contentsline {section}{\numberline {4.14}\textcolor {Chapter }{Octahedral manifolds}}{62}{section.4.14}% +\contentsline {section}{\numberline {4.15}\textcolor {Chapter }{Dodecahedral manifolds}}{62}{section.4.15}% +\contentsline {section}{\numberline {4.16}\textcolor {Chapter }{Prism manifolds}}{63}{section.4.16}% +\contentsline {section}{\numberline {4.17}\textcolor {Chapter }{Bipyramid manifolds}}{64}{section.4.17}% +\contentsline {chapter}{\numberline {5}\textcolor {Chapter }{Topological data analysis}}{65}{chapter.5}% +\contentsline {section}{\numberline {5.1}\textcolor {Chapter }{Persistent homology }}{65}{section.5.1}% +\contentsline {subsection}{\numberline {5.1.1}\textcolor {Chapter }{Background to the data}}{66}{subsection.5.1.1}% +\contentsline {section}{\numberline {5.2}\textcolor {Chapter }{Mapper clustering}}{66}{section.5.2}% +\contentsline {subsection}{\numberline {5.2.1}\textcolor {Chapter }{Background to the data}}{67}{subsection.5.2.1}% +\contentsline {section}{\numberline {5.3}\textcolor {Chapter }{Some tools for handling pure complexes}}{67}{section.5.3}% +\contentsline {section}{\numberline {5.4}\textcolor {Chapter }{Digital image analysis and persistent homology}}{68}{section.5.4}% +\contentsline {subsection}{\numberline {5.4.1}\textcolor {Chapter }{Naive example of image segmentation by automatic thresholding}}{68}{subsection.5.4.1}% +\contentsline {subsection}{\numberline {5.4.2}\textcolor {Chapter }{Refining the filtration}}{69}{subsection.5.4.2}% +\contentsline {subsection}{\numberline {5.4.3}\textcolor {Chapter }{Background to the data}}{69}{subsection.5.4.3}% +\contentsline {section}{\numberline {5.5}\textcolor {Chapter }{A second example of digital image segmentation}}{69}{section.5.5}% +\contentsline {section}{\numberline {5.6}\textcolor {Chapter }{A third example of digital image segmentation}}{70}{section.5.6}% +\contentsline {section}{\numberline {5.7}\textcolor {Chapter }{Naive example of digital image contour extraction}}{71}{section.5.7}% +\contentsline {section}{\numberline {5.8}\textcolor {Chapter }{Alternative approaches to computing persistent homology}}{72}{section.5.8}% +\contentsline {subsection}{\numberline {5.8.1}\textcolor {Chapter }{Non-trivial cup product}}{73}{subsection.5.8.1}% +\contentsline {subsection}{\numberline {5.8.2}\textcolor {Chapter }{Explicit homology generators}}{73}{subsection.5.8.2}% +\contentsline {section}{\numberline {5.9}\textcolor {Chapter }{Knotted proteins}}{74}{section.5.9}% +\contentsline {section}{\numberline {5.10}\textcolor {Chapter }{Random simplicial complexes}}{75}{section.5.10}% +\contentsline {section}{\numberline {5.11}\textcolor {Chapter }{Computing homology of a clique complex (Vietoris-Rips complex) }}{77}{section.5.11}% +\contentsline {chapter}{\numberline {6}\textcolor {Chapter }{Group theoretic computations}}{79}{chapter.6}% +\contentsline {section}{\numberline {6.1}\textcolor {Chapter }{Third homotopy group of a supsension of an Eilenberg-MacLane space }}{79}{section.6.1}% +\contentsline {section}{\numberline {6.2}\textcolor {Chapter }{Representations of knot quandles}}{79}{section.6.2}% +\contentsline {section}{\numberline {6.3}\textcolor {Chapter }{Identifying knots}}{80}{section.6.3}% +\contentsline {section}{\numberline {6.4}\textcolor {Chapter }{Aspherical $2$-complexes}}{80}{section.6.4}% +\contentsline {section}{\numberline {6.5}\textcolor {Chapter }{Group presentations and homotopical syzygies}}{80}{section.6.5}% +\contentsline {section}{\numberline {6.6}\textcolor {Chapter }{Bogomolov multiplier}}{82}{section.6.6}% +\contentsline {section}{\numberline {6.7}\textcolor {Chapter }{Second group cohomology and group extensions}}{82}{section.6.7}% +\contentsline {section}{\numberline {6.8}\textcolor {Chapter }{Second group cohomology and cocyclic Hadamard matrices}}{85}{section.6.8}% +\contentsline {section}{\numberline {6.9}\textcolor {Chapter }{Third group cohomology and homotopy $2$-types}}{85}{section.6.9}% +\contentsline {chapter}{\numberline {7}\textcolor {Chapter }{Cohomology of groups (and Lie Algebras)}}{87}{chapter.7}% +\contentsline {section}{\numberline {7.1}\textcolor {Chapter }{Finite groups }}{87}{section.7.1}% +\contentsline {subsection}{\numberline {7.1.1}\textcolor {Chapter }{Naive homology computation for a very small group}}{87}{subsection.7.1.1}% +\contentsline {subsection}{\numberline {7.1.2}\textcolor {Chapter }{A more efficient homology computation}}{87}{subsection.7.1.2}% +\contentsline {subsection}{\numberline {7.1.3}\textcolor {Chapter }{Computation of an induced homology homomorphism}}{88}{subsection.7.1.3}% +\contentsline {subsection}{\numberline {7.1.4}\textcolor {Chapter }{Some other finite group homology computations}}{88}{subsection.7.1.4}% +\contentsline {section}{\numberline {7.2}\textcolor {Chapter }{Nilpotent groups}}{90}{section.7.2}% +\contentsline {section}{\numberline {7.3}\textcolor {Chapter }{Crystallographic and Almost Crystallographic groups}}{90}{section.7.3}% +\contentsline {section}{\numberline {7.4}\textcolor {Chapter }{Arithmetic groups}}{91}{section.7.4}% +\contentsline {section}{\numberline {7.5}\textcolor {Chapter }{Artin groups}}{91}{section.7.5}% +\contentsline {section}{\numberline {7.6}\textcolor {Chapter }{Graphs of groups}}{91}{section.7.6}% +\contentsline {section}{\numberline {7.7}\textcolor {Chapter }{Lie algebra homology and free nilpotent groups}}{92}{section.7.7}% +\contentsline {section}{\numberline {7.8}\textcolor {Chapter }{Cohomology with coefficients in a module}}{93}{section.7.8}% +\contentsline {section}{\numberline {7.9}\textcolor {Chapter }{Cohomology as a functor of the first variable}}{95}{section.7.9}% +\contentsline {section}{\numberline {7.10}\textcolor {Chapter }{Cohomology as a functor of the second variable and the long exact coefficient sequence}}{96}{section.7.10}% +\contentsline {section}{\numberline {7.11}\textcolor {Chapter }{Transfer Homomorphism}}{97}{section.7.11}% +\contentsline {section}{\numberline {7.12}\textcolor {Chapter }{Cohomology rings of finite fundamental groups of 3-manifolds }}{97}{section.7.12}% +\contentsline {section}{\numberline {7.13}\textcolor {Chapter }{Explicit cocycles }}{99}{section.7.13}% +\contentsline {section}{\numberline {7.14}\textcolor {Chapter }{Quillen's complex and the $p$-part of homology }}{102}{section.7.14}% +\contentsline {section}{\numberline {7.15}\textcolor {Chapter }{Homology of a Lie algebra with coefficients in a module}}{104}{section.7.15}% +\contentsline {chapter}{\numberline {8}\textcolor {Chapter }{Cohomology rings and Steenrod operations for finite groups}}{106}{chapter.8}% +\contentsline {section}{\numberline {8.1}\textcolor {Chapter }{Mod-$p$ cohomology rings of finite groups}}{106}{section.8.1}% +\contentsline {subsection}{\numberline {8.1.1}\textcolor {Chapter }{Ring presentations (for the commutative $p=2$ case)}}{107}{subsection.8.1.1}% +\contentsline {section}{\numberline {8.2}\textcolor {Chapter }{Functorial ring homomorphisms in Mod-$p$ cohomology}}{107}{section.8.2}% +\contentsline {subsection}{\numberline {8.2.1}\textcolor {Chapter }{Testing homomorphism properties}}{107}{subsection.8.2.1}% +\contentsline {subsection}{\numberline {8.2.2}\textcolor {Chapter }{Testing functorial properties}}{108}{subsection.8.2.2}% +\contentsline {subsection}{\numberline {8.2.3}\textcolor {Chapter }{Computing with larger groups}}{108}{subsection.8.2.3}% +\contentsline {section}{\numberline {8.3}\textcolor {Chapter }{Cohomology rings of finite $2$-groups}}{109}{section.8.3}% +\contentsline {section}{\numberline {8.4}\textcolor {Chapter }{Steenrod operations for finite $2$-groups}}{110}{section.8.4}% +\contentsline {section}{\numberline {8.5}\textcolor {Chapter }{Steenrod operations on the classifying space of a finite $p$-group}}{111}{section.8.5}% +\contentsline {chapter}{\numberline {9}\textcolor {Chapter }{Bredon homology}}{112}{chapter.9}% +\contentsline {section}{\numberline {9.1}\textcolor {Chapter }{Davis complex}}{112}{section.9.1}% +\contentsline {section}{\numberline {9.2}\textcolor {Chapter }{Arithmetic groups}}{112}{section.9.2}% +\contentsline {section}{\numberline {9.3}\textcolor {Chapter }{Crystallographic groups}}{113}{section.9.3}% +\contentsline {chapter}{\numberline {10}\textcolor {Chapter }{Chain Complexes}}{114}{chapter.10}% +\contentsline {section}{\numberline {10.1}\textcolor {Chapter }{Chain complex of a simplicial complex and simplicial pair}}{114}{section.10.1}% +\contentsline {section}{\numberline {10.2}\textcolor {Chapter }{Chain complex of a cubical complex and cubical pair}}{115}{section.10.2}% +\contentsline {section}{\numberline {10.3}\textcolor {Chapter }{Chain complex of a regular CW-complex}}{116}{section.10.3}% +\contentsline {section}{\numberline {10.4}\textcolor {Chapter }{Chain Maps of simplicial and regular CW maps}}{117}{section.10.4}% +\contentsline {section}{\numberline {10.5}\textcolor {Chapter }{Constructions for chain complexes}}{117}{section.10.5}% +\contentsline {section}{\numberline {10.6}\textcolor {Chapter }{Filtered chain complexes}}{118}{section.10.6}% +\contentsline {section}{\numberline {10.7}\textcolor {Chapter }{Sparse chain complexes}}{119}{section.10.7}% +\contentsline {chapter}{\numberline {11}\textcolor {Chapter }{Resolutions}}{121}{chapter.11}% +\contentsline {section}{\numberline {11.1}\textcolor {Chapter }{Resolutions for small finite groups}}{121}{section.11.1}% +\contentsline {section}{\numberline {11.2}\textcolor {Chapter }{Resolutions for very small finite groups}}{121}{section.11.2}% +\contentsline {section}{\numberline {11.3}\textcolor {Chapter }{Resolutions for finite groups acting on orbit polytopes}}{123}{section.11.3}% +\contentsline {section}{\numberline {11.4}\textcolor {Chapter }{Minimal resolutions for finite $p$-groups over $\mathbb F_p$}}{124}{section.11.4}% +\contentsline {section}{\numberline {11.5}\textcolor {Chapter }{Resolutions for abelian groups}}{124}{section.11.5}% +\contentsline {section}{\numberline {11.6}\textcolor {Chapter }{Resolutions for nilpotent groups}}{125}{section.11.6}% +\contentsline {section}{\numberline {11.7}\textcolor {Chapter }{Resolutions for groups with subnormal series}}{126}{section.11.7}% +\contentsline {section}{\numberline {11.8}\textcolor {Chapter }{Resolutions for groups with normal series}}{126}{section.11.8}% +\contentsline {section}{\numberline {11.9}\textcolor {Chapter }{Resolutions for polycyclic (almost) crystallographic groups }}{126}{section.11.9}% +\contentsline {section}{\numberline {11.10}\textcolor {Chapter }{Resolutions for Bieberbach groups }}{127}{section.11.10}% +\contentsline {section}{\numberline {11.11}\textcolor {Chapter }{Resolutions for arbitrary crystallographic groups}}{128}{section.11.11}% +\contentsline {section}{\numberline {11.12}\textcolor {Chapter }{Resolutions for crystallographic groups admitting cubical fundamental domain}}{128}{section.11.12}% +\contentsline {section}{\numberline {11.13}\textcolor {Chapter }{Resolutions for Coxeter groups }}{129}{section.11.13}% +\contentsline {section}{\numberline {11.14}\textcolor {Chapter }{Resolutions for Artin groups }}{129}{section.11.14}% +\contentsline {section}{\numberline {11.15}\textcolor {Chapter }{Resolutions for $G=SL_2(\mathbb Z[1/m])$}}{130}{section.11.15}% +\contentsline {section}{\numberline {11.16}\textcolor {Chapter }{Resolutions for selected groups $G=SL_2( {\mathcal O}(\mathbb Q(\sqrt {d}) )$}}{130}{section.11.16}% +\contentsline {section}{\numberline {11.17}\textcolor {Chapter }{Resolutions for selected groups $G=PSL_2( {\mathcal O}(\mathbb Q(\sqrt {d}) )$}}{130}{section.11.17}% +\contentsline {section}{\numberline {11.18}\textcolor {Chapter }{Resolutions for a few higher-dimensional arithmetic groups }}{131}{section.11.18}% +\contentsline {section}{\numberline {11.19}\textcolor {Chapter }{Resolutions for finite-index subgroups }}{131}{section.11.19}% +\contentsline {section}{\numberline {11.20}\textcolor {Chapter }{Simplifying resolutions }}{132}{section.11.20}% +\contentsline {section}{\numberline {11.21}\textcolor {Chapter }{Resolutions for graphs of groups and for groups with aspherical presentations }}{132}{section.11.21}% +\contentsline {section}{\numberline {11.22}\textcolor {Chapter }{Resolutions for $\mathbb FG$-modules }}{133}{section.11.22}% +\contentsline {chapter}{\numberline {12}\textcolor {Chapter }{Simplicial groups}}{134}{chapter.12}% +\contentsline {section}{\numberline {12.1}\textcolor {Chapter }{Crossed modules}}{134}{section.12.1}% +\contentsline {section}{\numberline {12.2}\textcolor {Chapter }{Eilenberg-MacLane spaces as simplicial groups (not recommended)}}{135}{section.12.2}% +\contentsline {section}{\numberline {12.3}\textcolor {Chapter }{Eilenberg-MacLane spaces as simplicial free abelian groups (recommended)}}{135}{section.12.3}% +\contentsline {section}{\numberline {12.4}\textcolor {Chapter }{Elementary theoretical information on $H^\ast (K(\pi ,n),\mathbb Z)$}}{137}{section.12.4}% +\contentsline {section}{\numberline {12.5}\textcolor {Chapter }{The first three non-trivial homotopy groups of spheres}}{138}{section.12.5}% +\contentsline {section}{\numberline {12.6}\textcolor {Chapter }{The first two non-trivial homotopy groups of the suspension and double suspension of a $K(G,1)$}}{139}{section.12.6}% +\contentsline {section}{\numberline {12.7}\textcolor {Chapter }{Postnikov towers and $\pi _5(S^3)$ }}{139}{section.12.7}% +\contentsline {section}{\numberline {12.8}\textcolor {Chapter }{Towards $\pi _4(\Sigma K(G,1))$ }}{141}{section.12.8}% +\contentsline {section}{\numberline {12.9}\textcolor {Chapter }{Enumerating homotopy 2-types}}{142}{section.12.9}% +\contentsline {section}{\numberline {12.10}\textcolor {Chapter }{Identifying cat$^1$-groups of low order}}{143}{section.12.10}% +\contentsline {section}{\numberline {12.11}\textcolor {Chapter }{Identifying crossed modules of low order}}{144}{section.12.11}% +\contentsline {chapter}{\numberline {13}\textcolor {Chapter }{Congruence Subgroups, Cuspidal Cohomology and Hecke Operators}}{146}{chapter.13}% +\contentsline {section}{\numberline {13.1}\textcolor {Chapter }{Eichler-Shimura isomorphism}}{146}{section.13.1}% +\contentsline {section}{\numberline {13.2}\textcolor {Chapter }{Generators for $SL_2(\mathbb Z)$ and the cubic tree}}{147}{section.13.2}% +\contentsline {section}{\numberline {13.3}\textcolor {Chapter }{One-dimensional fundamental domains and generators for congruence subgroups}}{148}{section.13.3}% +\contentsline {section}{\numberline {13.4}\textcolor {Chapter }{Cohomology of congruence subgroups}}{149}{section.13.4}% +\contentsline {subsection}{\numberline {13.4.1}\textcolor {Chapter }{Cohomology with rational coefficients}}{151}{subsection.13.4.1}% +\contentsline {section}{\numberline {13.5}\textcolor {Chapter }{Cuspidal cohomology}}{151}{section.13.5}% +\contentsline {section}{\numberline {13.6}\textcolor {Chapter }{Hecke operators on forms of weight 2}}{153}{section.13.6}% +\contentsline {section}{\numberline {13.7}\textcolor {Chapter }{Hecke operators on forms of weight $ \ge 2$}}{154}{section.13.7}% +\contentsline {section}{\numberline {13.8}\textcolor {Chapter }{Reconstructing modular forms from cohomology computations}}{154}{section.13.8}% +\contentsline {section}{\numberline {13.9}\textcolor {Chapter }{The Picard group}}{156}{section.13.9}% +\contentsline {section}{\numberline {13.10}\textcolor {Chapter }{Bianchi groups}}{158}{section.13.10}% +\contentsline {section}{\numberline {13.11}\textcolor {Chapter }{Some other infinite matrix groups}}{159}{section.13.11}% +\contentsline {section}{\numberline {13.12}\textcolor {Chapter }{Ideals and finite quotient groups}}{161}{section.13.12}% +\contentsline {section}{\numberline {13.13}\textcolor {Chapter }{Congruence subgroups for ideals}}{163}{section.13.13}% +\contentsline {section}{\numberline {13.14}\textcolor {Chapter }{First homology}}{164}{section.13.14}% +\contentsline {chapter}{\numberline {14}\textcolor {Chapter }{Parallel computation}}{166}{chapter.14}% +\contentsline {section}{\numberline {14.1}\textcolor {Chapter }{An embarassingly parallel computation}}{166}{section.14.1}% +\contentsline {section}{\numberline {14.2}\textcolor {Chapter }{An non-embarassingly parallel computation}}{166}{section.14.2}% +\contentsline {chapter}{\numberline {15}\textcolor {Chapter }{Regular CW-structure on knots (written by Kelvin Killeen)}}{168}{chapter.15}% +\contentsline {section}{\numberline {15.1}\textcolor {Chapter }{Knot complements in the 3-ball}}{168}{section.15.1}% +\contentsline {section}{\numberline {15.2}\textcolor {Chapter }{Tubular neighbourhoods}}{169}{section.15.2}% +\contentsline {section}{\numberline {15.3}\textcolor {Chapter }{Knotted surface complements in the 4-ball}}{172}{section.15.3}% +\contentsline {chapter}{References}{181}{chapter*.2}% diff --git a/tutorial/HapTutorial.xml b/tutorial/HapTutorial.xml index d02d2dc4..dfa06c36 100644 --- a/tutorial/HapTutorial.xml +++ b/tutorial/HapTutorial.xml @@ -6,8 +6,8 @@ -A short HAP tutorial -(../www/SideLinks/About/aboutContents.htmlA more comprehensive tutorial is available here
+A newer HAP tutorial +(../www/SideLinks/About/aboutContents.htmlAn older tutorial is available here
and
https://global.oup.com/academic/product/an-invitation-to-computational-homotopy-9780198832980A related book is available here
and
diff --git a/tutorial/chap0.html b/tutorial/chap0.html index 3b3bf066..0fb01c5e 100644 --- a/tutorial/chap0.html +++ b/tutorial/chap0.html @@ -23,10 +23,10 @@

-

A short HAP tutorial

+

A newer HAP tutorial

-

(A more comprehensive tutorial is available here
and
A related book is available here
and
The HAP home page is here)

+

(An older tutorial is available here
and
A related book is available here
and
The HAP home page is here)

Graham Ellis @@ -272,7 +272,7 @@

Contents

-
7 Cohomology of groups +
7 Cohomology of groups (and Lie Algebras)
 7.1 Finite groups
@@ -325,6 +325,9 @@

Contents

 7.14 Quillen's complex and the p-part of homology
+
 7.15 Homology of a Lie algebra with coefficients in a module + +
8 Cohomology rings and Steenrod operations for finite groups
 8.1 Mod-p cohomology rings of finite groups diff --git a/tutorial/chap0.txt b/tutorial/chap0.txt index 8aef4847..cc73738f 100644 --- a/tutorial/chap0.txt +++ b/tutorial/chap0.txt @@ -1,9 +1,9 @@ - A short HAP tutorial + A newer HAP tutorial - (A more comprehensive tutorial is available here + (An older tutorial is available here (../www/SideLinks/About/aboutContents.html) and A related book is available here ( @@ -105,7 +105,7 @@ 6.7 Second group cohomology and group extensions 6.8 Second group cohomology and cocyclic Hadamard matrices 6.9 Third group cohomology and homotopy 2-types - 7 Cohomology of groups + 7 Cohomology of groups (and Lie Algebras) 7.1 Finite groups 7.1-1 Naive homology computation for a very small group 7.1-2 A more efficient homology computation @@ -125,6 +125,7 @@ 7.12 Cohomology rings of finite fundamental groups of 3-manifolds 7.13 Explicit cocycles 7.14 Quillen's complex and the p-part of homology + 7.15 Homology of a Lie algebra with coefficients in a module 8 Cohomology rings and Steenrod operations for finite groups 8.1 Mod-p cohomology rings of finite groups 8.1-1 Ring presentations (for the commutative p=2 case) diff --git a/tutorial/chap0_mj.html b/tutorial/chap0_mj.html index beada501..58f573d4 100644 --- a/tutorial/chap0_mj.html +++ b/tutorial/chap0_mj.html @@ -26,10 +26,10 @@

-

A short HAP tutorial

+

A newer HAP tutorial

-

(A more comprehensive tutorial is available here
and
A related book is available here
and
The HAP home page is here)

+

(An older tutorial is available here
and
A related book is available here
and
The HAP home page is here)

Graham Ellis @@ -275,7 +275,7 @@

Contents
-
7 Cohomology of groups +
7 Cohomology of groups (and Lie Algebras)
 7.1 Finite groups
@@ -328,6 +328,9 @@

Contents

 7.14 Quillen's complex and the \(p\)-part of homology
+
 7.15 Homology of a Lie algebra with coefficients in a module + +
8 Cohomology rings and Steenrod operations for finite groups
 8.1 Mod-\(p\) cohomology rings of finite groups diff --git a/tutorial/chap12.html b/tutorial/chap12.html index e0661b67..588d5c79 100644 --- a/tutorial/chap12.html +++ b/tutorial/chap12.html @@ -76,7 +76,7 @@

12.1 Crossed modules

for g∈ G, m,m'∈ M.

-

A crossed module ∂: M→ G is equivalent to a cat^1-group (H,s,t) (see 6.8) where H=M ⋊ G, s(m,g) = (1,g), t(m,g)=(1,(∂ m)g). A cat^1-group is, in turn, equivalent to a simplicial group with Moore complex has length 1. The simplicial group is constructed by considering the cat^1-group as a category and taking its nerve. Alternatively, the simplicial group can be constructed by viewing the crossed module as a crossed complex and using a nonabelian version of the Dold-Kan theorem.

+

A crossed module ∂: M→ G is equivalent to a cat^1-group (H,s,t) (see 6.9) where H=M ⋊ G, s(m,g) = (1,g), t(m,g)=(1,(∂ m)g). A cat^1-group is, in turn, equivalent to a simplicial group with Moore complex has length 1. The simplicial group is constructed by considering the cat^1-group as a category and taking its nerve. Alternatively, the simplicial group can be constructed by viewing the crossed module as a crossed complex and using a nonabelian version of the Dold-Kan theorem.

The following example concerns the crossed module

@@ -884,7 +884,7 @@

12.8 Towards π_4(Σ K(G,1))<

12.9 Enumerating homotopy 2-types

-

A 2-type is a CW-complex X whose homotopy groups are trivial in dimensions n=0 and n>2. As explained in 6.8 the homotopy type of such a space can be captured algebraically by a cat^1-group G. Let X, Y be 2-tytpes represented by cat^1-groups G, H. If X and Y are homotopy equivalent then there exists a sequence of morphisms of cat^1-groups

+

A 2-type is a CW-complex X whose homotopy groups are trivial in dimensions n=0 and n>2. As explained in 6.9 the homotopy type of such a space can be captured algebraically by a cat^1-group G. Let X, Y be 2-tytpes represented by cat^1-groups G, H. If X and Y are homotopy equivalent then there exists a sequence of morphisms of cat^1-groups

G \rightarrow K_1 \rightarrow K_2 \leftarrow K_3 \rightarrow \cdots \rightarrow K_n \leftarrow H

diff --git a/tutorial/chap12.txt b/tutorial/chap12.txt index f6953023..d060f295 100644 --- a/tutorial/chap12.txt +++ b/tutorial/chap12.txt @@ -13,7 +13,7 @@ for g∈ G, m,m'∈ M. - A crossed module ∂: M→ G is equivalent to a cat^1-group (H,s,t) (see 6.8) + A crossed module ∂: M→ G is equivalent to a cat^1-group (H,s,t) (see 6.9) where H=M ⋊ G, s(m,g) = (1,g), t(m,g)=(1,(∂ m)g). A cat^1-group is, in turn, equivalent to a simplicial group with Moore complex has length 1. The simplicial group is constructed by considering the cat^1-group as a category @@ -533,7 +533,7 @@ 12.9 Enumerating homotopy 2-types A 2-type is a CW-complex X whose homotopy groups are trivial in dimensions - n=0 and n>2. As explained in 6.8 the homotopy type of such a space can be + n=0 and n>2. As explained in 6.9 the homotopy type of such a space can be captured algebraically by a cat^1-group G. Let X, Y be 2-tytpes represented by cat^1-groups G, H. If X and Y are homotopy equivalent then there exists a sequence of morphisms of cat^1-groups diff --git a/tutorial/chap12_mj.html b/tutorial/chap12_mj.html index 97b889e5..ef4f960f 100644 --- a/tutorial/chap12_mj.html +++ b/tutorial/chap12_mj.html @@ -79,7 +79,7 @@

12.1 Crossed modules

for \(g\in G\), \(m,m'\in M\).

-

A crossed module \(\partial\colon M\rightarrow G\) is equivalent to a cat\(^1\)-group \((H,s,t)\) (see 6.8) where \(H=M \rtimes G\), \(s(m,g) = (1,g)\), \(t(m,g)=(1,(\partial m)g)\). A cat\(^1\)-group is, in turn, equivalent to a simplicial group with Moore complex has length \(1\). The simplicial group is constructed by considering the cat\(^1\)-group as a category and taking its nerve. Alternatively, the simplicial group can be constructed by viewing the crossed module as a crossed complex and using a nonabelian version of the Dold-Kan theorem.

+

A crossed module \(\partial\colon M\rightarrow G\) is equivalent to a cat\(^1\)-group \((H,s,t)\) (see 6.9) where \(H=M \rtimes G\), \(s(m,g) = (1,g)\), \(t(m,g)=(1,(\partial m)g)\). A cat\(^1\)-group is, in turn, equivalent to a simplicial group with Moore complex has length \(1\). The simplicial group is constructed by considering the cat\(^1\)-group as a category and taking its nerve. Alternatively, the simplicial group can be constructed by viewing the crossed module as a crossed complex and using a nonabelian version of the Dold-Kan theorem.

The following example concerns the crossed module

@@ -887,7 +887,7 @@

12.8 Towards \(\pi_4(\Sigma K

12.9 Enumerating homotopy 2-types

-

A 2-type is a CW-complex \(X\) whose homotopy groups are trivial in dimensions \(n=0 \) and \(n>2\). As explained in 6.8 the homotopy type of such a space can be captured algebraically by a cat\(^1\)-group \(G\). Let \(X\), \(Y\) be \(2\)-tytpes represented by cat\(^1\)-groups \(G\), \(H\). If \(X\) and \(Y\) are homotopy equivalent then there exists a sequence of morphisms of cat\(^1\)-groups

+

A 2-type is a CW-complex \(X\) whose homotopy groups are trivial in dimensions \(n=0 \) and \(n>2\). As explained in 6.9 the homotopy type of such a space can be captured algebraically by a cat\(^1\)-group \(G\). Let \(X\), \(Y\) be \(2\)-tytpes represented by cat\(^1\)-groups \(G\), \(H\). If \(X\) and \(Y\) are homotopy equivalent then there exists a sequence of morphisms of cat\(^1\)-groups

\[G \rightarrow K_1 \rightarrow K_2 \leftarrow K_3 \rightarrow \cdots \rightarrow K_n \leftarrow H\]

diff --git a/tutorial/chap5.html b/tutorial/chap5.html index b28e07df..862d7664 100644 --- a/tutorial/chap5.html +++ b/tutorial/chap5.html @@ -223,7 +223,7 @@

5.3 Some tools for handling pure complexes

  • ContractedComplex(M, optional subcomplex of M)

    • -

  • ExpandedComplex(M, optional subcomplex of M)

    +

  • ExpandedComplex(M, optional supercomplex of M)

  • PureComplexUnion(M,N)

    @@ -412,14 +412,15 @@

    5.6 A third example of digital image segmentationgap> BettiNumber(FiltrationTerm(T,12),1); 0 -gap> #Confirmation that 11-th filtration term has one hole. +gap> #Confirmation that 11-th filtration term has one hole and the 12-th term is contractible. gap> C:=FiltrationTerm(T,11);; gap> for n in Reversed([1..10]) do > C:=ContractedComplex(C,FiltrationTerm(T,n)); > od; gap> C:=PureComplexBoundary(PureComplexThickened(C));; gap> H:=HomotopyEquivalentMinimalPureCubicalSubcomplex(FiltrationTerm(T,12),C);; -gap> Display(PureComplexBoundary(H)); +gap> B:=ContractedComplex(PureComplexBoundary(H));; +gap> Display(B);

    • diff --git a/tutorial/chap5.txt b/tutorial/chap5.txt index 74ad34bd..673cda1d 100644 --- a/tutorial/chap5.txt +++ b/tutorial/chap5.txt @@ -175,7 +175,7 @@  ContractedComplex(M, optional subcomplex of M) -  ExpandedComplex(M, optional subcomplex of M) +  ExpandedComplex(M, optional supercomplex of M)  PureComplexUnion(M,N) @@ -356,14 +356,15 @@ 1 gap> BettiNumber(FiltrationTerm(T,12),1); 0 - gap> #Confirmation that 11-th filtration term has one hole. + gap> #Confirmation that 11-th filtration term has one hole and the 12-th term is contractible. gap> C:=FiltrationTerm(T,11);; gap> for n in Reversed([1..10]) do > C:=ContractedComplex(C,FiltrationTerm(T,n)); > od; gap> C:=PureComplexBoundary(PureComplexThickened(C));; gap> H:=HomotopyEquivalentMinimalPureCubicalSubcomplex(FiltrationTerm(T,12),C);; - gap> Display(PureComplexBoundary(H)); + gap> B:=ContractedComplex(PureComplexBoundary(H));; + gap> Display(B);   diff --git a/tutorial/chap5_mj.html b/tutorial/chap5_mj.html index 866db233..feee6076 100644 --- a/tutorial/chap5_mj.html +++ b/tutorial/chap5_mj.html @@ -226,7 +226,7 @@

    5.3 Some tools for handling pure complexes

  • ContractedComplex(M, optional subcomplex of M)

    • -

  • ExpandedComplex(M, optional subcomplex of M)

    +

  • ExpandedComplex(M, optional supercomplex of M)

  • PureComplexUnion(M,N)

    @@ -415,14 +415,15 @@

    5.6 A third example of digital image segmentationgap> BettiNumber(FiltrationTerm(T,12),1); 0 -gap> #Confirmation that 11-th filtration term has one hole. +gap> #Confirmation that 11-th filtration term has one hole and the 12-th term is contractible. gap> C:=FiltrationTerm(T,11);; gap> for n in Reversed([1..10]) do > C:=ContractedComplex(C,FiltrationTerm(T,n)); > od; gap> C:=PureComplexBoundary(PureComplexThickened(C));; gap> H:=HomotopyEquivalentMinimalPureCubicalSubcomplex(FiltrationTerm(T,12),C);; -gap> Display(PureComplexBoundary(H)); +gap> B:=ContractedComplex(PureComplexBoundary(H));; +gap> Display(B);

    • diff --git a/tutorial/chap7.html b/tutorial/chap7.html index b41a7718..28e529b4 100644 --- a/tutorial/chap7.html +++ b/tutorial/chap7.html @@ -5,7 +5,7 @@ -GAP (HAP commands) - Chapter 7: Cohomology of groups +GAP (HAP commands) - Chapter 7: Cohomology of groups (and Lie Algebras) @@ -20,8 +20,8 @@
     [Top of Book]  [Contents]   [Previous Chapter]   [Next Chapter] 
    -

    -
    7 Cohomology of groups +

    +
    7 Cohomology of groups (and Lie Algebras)
     7.1 Finite groups
    @@ -74,9 +74,12 @@
     7.14 Quillen's complex and the p-part of homology
    +
     7.15 Homology of a Lie algebra with coefficients in a module + +
    -

    7 Cohomology of groups

    +

    7 Cohomology of groups (and Lie Algebras)

    @@ -854,10 +857,10 @@

    7.13 Explicit cocycles

  • Let G be a finite group and k a field of characteristic 0. The group algebra k(G), and the algebra F(G) of functions d_g: G→ k, h→ d_g,h, are both Hopf algebras. The tensor product F(G) ⊗ k(G) also admits a Hopf algebra structure known as the quantum double D(G). A twisted quantum double D_f(G) was introduced by R. Dijkraaf, V. Pasquier & P. Roche [DPR91]. The twisted double is a quasi-Hopf algebra depending on a 3-cocycle f: G× G× G→ k. The multiplication is given by (d_g ⊗ x)(d_h ⊗ y) = d_gx,xhβ_g(x,y)(d_g ⊗ xy) where β_a is defined by β_a(h,g) = f(a,h,g) f(h,h^-1ah,g)^-1 f(h,g,(hg)^-1ahg) . Although the algebraic structure of D_f(G) depends very much on the particular 3-cocycle f, representation-theoretic properties of D_f(G) depend only on the cohomology class of f.

    • -

  • An explicit 2-cocycle f: G× G→ A is needed to construct the multiplication (a,g)(a',g') = (a + g⋅ a' + f(g,g'), gg') in the extension a group G by a ZG-module A determined by the cohomology class of f in H^2(G,A). See 6.6.

    +

  • An explicit 2-cocycle f: G× G→ A is needed to construct the multiplication (a,g)(a',g') = (a + g⋅ a' + f(g,g'), gg') in the extension a group G by a ZG-module A determined by the cohomology class of f in H^2(G,A). See 6.7.

    • -

  • In work on coding theory and Hadamard matrices a number of papers have investigated square matrices (a_ij) whose entries a_ij=f(g_i,g_j) are the values of a 2-cocycle f: G× G → Z_2 where G is a finite group acting trivially on Z_2. See for instance [Hor00] and 6.7.

    +

  • In work on coding theory and Hadamard matrices a number of papers have investigated square matrices (a_ij) whose entries a_ij=f(g_i,g_j) are the values of a 2-cocycle f: G× G → Z_2 where G is a finite group acting trivially on Z_2. See for instance [Hor00] and 6.8.

    • @@ -1101,6 +1104,28 @@

    7.14 Quillen's complex and the

    + +

    7.15 Homology of a Lie algebra with coefficients in a module

    + +

    Let A be the Lie algebra constructed from the associative algebra M^4× 4( Q) of all 4× 4 rational matrices. Let V be its adjoint module (with underlying vector space of dimension 16 and equal to that of A). The following commands compute H_4(A,V) = Q.

    + + +
    +gap> M:=FullMatrixAlgebra(Rationals,4);; 
+gap> A:=LieAlgebra(M);;
+gap> V:=AdjointModule(A);;
+gap> C:=ChevalleyEilenbergComplex(V,17);;
+gap> List([0..17],C!.dimension);
+[ 16, 256, 1920, 8960, 29120, 69888, 128128, 183040, 205920, 183040, 128128, 
+  69888, 29120, 8960, 1920, 256, 16, 0 ]
+gap> Homology(C,4);
+1
+
+
    + +

    Note that the eighth term C_8(V) in the Chevalley-Eilenberg complex C_∗(V) is a vector space of dimension 205920 and so it will take longer to compute the homology in degree 8.

    +
     [Top of Book]  [Contents]   [Previous Chapter]   [Next Chapter] 
    diff --git a/tutorial/chap7.txt b/tutorial/chap7.txt index b10555ee..e6aa1b0d 100644 --- a/tutorial/chap7.txt +++ b/tutorial/chap7.txt @@ -1,5 +1,5 @@ - 7 Cohomology of groups + 7 Cohomology of groups (and Lie Algebras) 7.1 Finite groups @@ -848,12 +848,12 @@  An explicit 2-cocycle f: G× G→ A is needed to construct the multiplication (a,g)(a',g') = (a + g⋅ a' + f(g,g'), gg') in the extension a group G by a ZG-module A determined by the cohomology - class of f in H^2(G,A). See 6.6. + class of f in H^2(G,A). See 6.7.  In work on coding theory and Hadamard matrices a number of papers have investigated square matrices (a_ij) whose entries a_ij=f(g_i,g_j) are the values of a 2-cocycle f: G× G → Z_2 where G is a finite group - acting trivially on Z_2. See for instance [Hor00] and 6.7. + acting trivially on Z_2. See for instance [Hor00] and 6.8. Given a ZG-resolution R_∗ (with contracting homotopy) and a ZG-module A one can use HAP commands to compute explicit standard n-cocycles f: G^n → A. @@ -1123,3 +1123,28 @@   + + 7.15 Homology of a Lie algebra with coefficients in a module + + Let A be the Lie algebra constructed from the associative algebra M^4× 4( Q) + of all 4× 4 rational matrices. Let V be its adjoint module (with underlying + vector space of dimension 16 and equal to that of A). The following commands + compute H_4(A,V) = Q. + +  Example  + gap> M:=FullMatrixAlgebra(Rationals,4);;  + gap> A:=LieAlgebra(M);; + gap> V:=AdjointModule(A);; + gap> C:=ChevalleyEilenbergComplex(V,17);; + gap> List([0..17],C!.dimension); + [ 16, 256, 1920, 8960, 29120, 69888, 128128, 183040, 205920, 183040, 128128,  +  69888, 29120, 8960, 1920, 256, 16, 0 ] + gap> Homology(C,4); + 1 +  +  + + Note that the eighth term C_8(V) in the Chevalley-Eilenberg complex C_∗(V) + is a vector space of dimension 205920 and so it will take longer to compute + the homology in degree 8. + diff --git a/tutorial/chap7_mj.html b/tutorial/chap7_mj.html index 980c7e17..bc231bc3 100644 --- a/tutorial/chap7_mj.html +++ b/tutorial/chap7_mj.html @@ -8,7 +8,7 @@ -GAP (HAP commands) - Chapter 7: Cohomology of groups +GAP (HAP commands) - Chapter 7: Cohomology of groups (and Lie Algebras) @@ -23,8 +23,8 @@
     [Top of Book]  [Contents]   [Previous Chapter]   [Next Chapter] 
    -

    -
    7 Cohomology of groups +

    +
    7 Cohomology of groups (and Lie Algebras)
     7.1 Finite groups
    @@ -77,9 +77,12 @@
     7.14 Quillen's complex and the \(p\)-part of homology
    +
     7.15 Homology of a Lie algebra with coefficients in a module + +
    -

    7 Cohomology of groups

    +

    7 Cohomology of groups (and Lie Algebras)

    @@ -857,10 +860,10 @@

    7.13 Explicit cocycles

  • Let \(G\) be a finite group and \(k\) a field of characteristic \(0\). The group algebra \(k(G)\), and the algebra \(F(G)\) of functions \(d_g\colon G\rightarrow k, h\rightarrow d_{g,h}\), are both Hopf algebras. The tensor product \(F(G) \otimes k(G)\) also admits a Hopf algebra structure known as the quantum double \(D(G)\). A twisted quantum double \(D_f(G)\) was introduced by R. Dijkraaf, V. Pasquier & P. Roche [DPR91]. The twisted double is a quasi-Hopf algebra depending on a \(3\)-cocycle \(f\colon G\times G\times G\rightarrow k\). The multiplication is given by \((d_g \otimes x)(d_h \otimes y) = d_{gx,xh}\beta_g(x,y)(d_g \otimes xy)\) where \(\beta_a \) is defined by \(\beta_a(h,g) = f(a,h,g) f(h,h^{-1}ah,g)^{-1} f(h,g,(hg)^{-1}ahg)\) . Although the algebraic structure of \(D_f(G)\) depends very much on the particular \(3\)-cocycle \(f\), representation-theoretic properties of \(D_f(G)\) depend only on the cohomology class of \(f\).

    • -

  • An explicit \(2\)-cocycle \(f\colon G\times G\rightarrow A\) is needed to construct the multiplication \((a,g)(a',g') = (a + g\cdot a' + f(g,g'), gg')\) in the extension a group \(G\) by a \(\mathbb ZG\)-module \(A\) determined by the cohomology class of \(f\) in \(H^2(G,A)\). See 6.6.

    +

  • An explicit \(2\)-cocycle \(f\colon G\times G\rightarrow A\) is needed to construct the multiplication \((a,g)(a',g') = (a + g\cdot a' + f(g,g'), gg')\) in the extension a group \(G\) by a \(\mathbb ZG\)-module \(A\) determined by the cohomology class of \(f\) in \(H^2(G,A)\). See 6.7.

    • -

  • In work on coding theory and Hadamard matrices a number of papers have investigated square matrices \((a_{ij})\) whose entries \(a_{ij}=f(g_i,g_j)\) are the values of a \(2\)-cocycle \(f\colon G\times G \rightarrow \mathbb Z_2\) where \(G\) is a finite group acting trivially on \(\mathbb Z_2\). See for instance [Hor00] and 6.7.

    +

  • In work on coding theory and Hadamard matrices a number of papers have investigated square matrices \((a_{ij})\) whose entries \(a_{ij}=f(g_i,g_j)\) are the values of a \(2\)-cocycle \(f\colon G\times G \rightarrow \mathbb Z_2\) where \(G\) is a finite group acting trivially on \(\mathbb Z_2\). See for instance [Hor00] and 6.8.

    • @@ -1104,6 +1107,28 @@

    7.14 Quillen's complex and the

    + +

    7.15 Homology of a Lie algebra with coefficients in a module

    + +

    Let \(A\) be the Lie algebra constructed from the associative algebra \(M^{4\times 4}(\mathbb Q)\) of all \(4\times 4\) rational matrices. Let \(V\) be its adjoint module (with underlying vector space of dimension \(16\) and equal to that of \(A\)). The following commands compute \(H_{4}(A,V) = \mathbb Q\).

    + + +
    +gap> M:=FullMatrixAlgebra(Rationals,4);; 
+gap> A:=LieAlgebra(M);;
+gap> V:=AdjointModule(A);;
+gap> C:=ChevalleyEilenbergComplex(V,17);;
+gap> List([0..17],C!.dimension);
+[ 16, 256, 1920, 8960, 29120, 69888, 128128, 183040, 205920, 183040, 128128, 
+  69888, 29120, 8960, 1920, 256, 16, 0 ]
+gap> Homology(C,4);
+1
+
+
    + +

    Note that the eighth term \(C_{8}(V)\) in the Chevalley-Eilenberg complex \(C_\ast(V)\) is a vector space of dimension \(205920\) and so it will take longer to compute the homology in degree \(8\).

    +
     [Top of Book]  [Contents]   [Previous Chapter]   [Next Chapter] 
    diff --git a/tutorial/images/3096final.png b/tutorial/images/3096final.png index 8335db25..9e4ee17f 100644 Binary files a/tutorial/images/3096final.png and b/tutorial/images/3096final.png differ diff --git a/tutorial/images/coinsbettizero.gif b/tutorial/images/coinsbettizero.gif index 6d8cd858..ae7c1ed4 100644 Binary files a/tutorial/images/coinsbettizero.gif and b/tutorial/images/coinsbettizero.gif differ diff --git a/tutorial/images/imbar0.gif b/tutorial/images/imbar0.gif index 4a8613cc..fd14ba8c 100644 Binary files a/tutorial/images/imbar0.gif and b/tutorial/images/imbar0.gif differ diff --git a/tutorial/images/imbar1.gif b/tutorial/images/imbar1.gif index a85859e2..e74ab330 100644 Binary files a/tutorial/images/imbar1.gif and b/tutorial/images/imbar1.gif differ diff --git a/tutorial/images/refinedbc.gif b/tutorial/images/refinedbc.gif new file mode 100644 index 00000000..7b0cae10 Binary files /dev/null and b/tutorial/images/refinedbc.gif differ diff --git 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"\033[1X\033[33X\033[0;-2YConstructing a CW-structure on a knot complement\033[133X\033[101X", "1.7", [ 1, 7, 0 ], 212, 9, - "constructing a cw-structure on a knot complement", "X7A15484C7E680AC9" ], [ "\033[1X\033[33X\033[0;-2YConstructing a regular CW-complex by attaching cells\033[133X\033[101X", "1.8", - [ 1, 8, 0 ], 247, 10, "constructing a regular cw-complex by attaching cells", "X829793717FB6DDCE" ], - [ "\033[1X\033[33X\033[0;-2YConstructing a regular CW-complex from its face lattice\033[133X\033[101X", "1.9", [ 1, 9, 0 ], 305, 11, "constructing a regular cw-complex from its face lattice", - "X7B7354E68025FC92" ], [ "\033[1X\033[33X\033[0;-2YCup products\033[133X\033[101X", "1.10", [ 1, 10, 0 ], 375, 12, "cup products", "X823FA6A9828FF473" ], - [ "\033[1X\033[33X\033[0;-2YIntersection forms of \033[22X4\033[122X\033[101X\027\033[1X\027-manifolds\033[133X\033[101X", "1.11", [ 1, 11, 0 ], 636, 17, "intersection forms of 4-manifolds", - "X7F9B01CF7EE1D2FC" ], [ 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"X85FB4CA987BC92CC" ], + [ + "\033[1X\033[33X\033[0;-2YSpun knots and the Satoh tube map\033[133X\033[10\ +1X", "3.2", [ 3, 2, 0 ], 81, 34, "spun knots and the satoh tube map", + "X7E5CC04E7E3CCDAD" ], + [ + "\033[1X\033[33X\033[0;-2YCohomology with local coefficients\033[133X\033[1\ +01X", "3.3", [ 3, 3, 0 ], 178, 36, "cohomology with local coefficients", + "X7C304A1C7EF0BA60" ], + [ + "\033[1X\033[33X\033[0;-2YDistinguishing between two non-homeomorphic homot\ +opy equivalent spaces\033[133X\033[101X", "3.4", [ 3, 4, 0 ], 218, 37, + "distinguishing between two non-homeomorphic homotopy equivalent spaces" + , "X7A4F34B780FA2CD5" ], + [ + "\033[1X\033[33X\033[0;-2YSecond homotopy groups of spaces with finite fund\ +amental group\033[133X\033[101X", "3.5", [ 3, 5, 0 ], 259, 37, + "second homotopy groups of spaces with finite fundamental group", + "X869FD75B84AAC7AD" ], + [ + "\033[1X\033[33X\033[0;-2YThird homotopy groups of simply connected spaces\\ +033[133X\033[101X", "3.6", [ 3, 6, 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yield homeomorphic manifolds", + "X7EC6B008878CC77E" ], + [ + "\033[1X\033[33X\033[0;-2YFinite fundamental groups of \033[22X3\033[122X\\ +033[101X\027\033[1X\027-manifolds\033[133X\033[101X", "4.8", [ 4, 8, 0 ], + 464, 49, "finite fundamental groups of 3-manifolds", + "X7B425A3280A2AF07" ], + [ "\033[1X\033[33X\033[0;-2YPoincare's cube manifolds\033[133X\033[101X", + "4.9", [ 4, 9, 0 ], 500, 50, "poincares cube manifolds", + "X78912D227D753167" ], + [ + "\033[1X\033[33X\033[0;-2YThere are at least 25 distinct cube manifolds\\ +033[133X\033[101X", "4.10", [ 4, 10, 0 ], 555, 51, + "there are at least 25 distinct cube manifolds", "X8761051F84C6CEC2" ], + [ "\033[1X\033[33X\033[0;-2YFace pairings for 25 distinct cube manifolds\033\ +[133X\033[101X", "4.10-1", [ 4, 10, 1 ], 672, 53, + "face pairings for 25 distinct cube manifolds", "X7D50795883E534A3" ], + [ "\033[1X\033[33X\033[0;-2YPlatonic cube manifolds\033[133X\033[101X", + "4.10-2", [ 4, 10, 2 ], 898, 57, "platonic cube manifolds", + 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81, "a more efficient homology computation", - "X838CEA3F850DFC82" ], [ "\033[1X\033[33X\033[0;-2YComputation of an induced homology homomorphism\033[133X\033[101X", "7.1-3", [ 7, 1, 3 ], 61, 82, - "computation of an induced homology homomorphism", "X842E93467AD09EC1" ], [ "\033[1X\033[33X\033[0;-2YSome other finite group homology computations\033[133X\033[101X", "7.1-4", [ 7, 1, 4 ], 89, - 82, "some other finite group homology computations", "X8754D2937E6FD7CE" ], [ "\033[1X\033[33X\033[0;-2YNilpotent groups\033[133X\033[101X", "7.2", [ 7, 2, 0 ], 208, 84, "nilpotent groups", - "X8463EF6A821FFB69" ], [ "\033[1X\033[33X\033[0;-2YCrystallographic and Almost Crystallographic groups\033[133X\033[101X", "7.3", [ 7, 3, 0 ], 227, 84, - "crystallographic and almost crystallographic groups", "X82E8FAC67BC16C01" ], [ "\033[1X\033[33X\033[0;-2YArithmetic groups\033[133X\033[101X", "7.4", [ 7, 4, 0 ], 256, 85, "arithmetic groups", - "X7AFFB32587D047FE" ], [ "\033[1X\033[33X\033[0;-2YArtin 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"\033[1X\033[33X\033[0;-2YComputation of an induced homology homomorphism\\ +033[133X\033[101X", "7.1-3", [ 7, 1, 3 ], 61, 88, + "computation of an induced homology homomorphism", "X842E93467AD09EC1" ] + , + [ + "\033[1X\033[33X\033[0;-2YSome other finite group homology computations\\ +033[133X\033[101X", "7.1-4", [ 7, 1, 4 ], 89, 88, + "some other finite group homology computations", "X8754D2937E6FD7CE" ], + [ "\033[1X\033[33X\033[0;-2YNilpotent groups\033[133X\033[101X", "7.2", + [ 7, 2, 0 ], 208, 90, "nilpotent groups", "X8463EF6A821FFB69" ], + [ + "\033[1X\033[33X\033[0;-2YCrystallographic and Almost Crystallographic grou\ +ps\033[133X\033[101X", "7.3", [ 7, 3, 0 ], 227, 90, + "crystallographic and almost crystallographic groups", + "X82E8FAC67BC16C01" ], + [ "\033[1X\033[33X\033[0;-2YArithmetic groups\033[133X\033[101X", "7.4", + [ 7, 4, 0 ], 256, 91, "arithmetic groups", "X7AFFB32587D047FE" ], + [ "\033[1X\033[33X\033[0;-2YArtin groups\033[133X\033[101X", "7.5", + [ 7, 5, 0 ], 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second variable and the long exact coeff\ +icient sequence", "X796731727A7EBE59" ], + [ "\033[1X\033[33X\033[0;-2YTransfer Homomorphism\033[133X\033[101X", + "7.11", [ 7, 11, 0 ], 655, 97, "transfer homomorphism", + "X80F6FD3E7C7E4E8D" ], + [ + "\033[1X\033[33X\033[0;-2YCohomology rings of finite fundamental groups of \ +3-manifolds\033[133X\033[101X", "7.12", [ 7, 12, 0 ], 688, 97, + "cohomology rings of finite fundamental groups of 3-manifolds", + "X79B1406C803FF178" ], + [ "\033[1X\033[33X\033[0;-2YExplicit cocycles\033[133X\033[101X", "7.13", + [ 7, 13, 0 ], 804, 99, "explicit cocycles", "X833A19F0791C3B06" ], + [ + "\033[1X\033[33X\033[0;-2YQuillen's complex and the \033[22Xp\033[122X\033[\ +101X\027\033[1X\027-part of homology\033[133X\033[101X", "7.14", + [ 7, 14, 0 ], 990, 102, "quillens complex and the p-part of homology", + "X7C5233E27D2D603E" ], + [ + "\033[1X\033[33X\033[0;-2YHomology of a Lie algebra with coefficients in a \ +module\033[133X\033[101X", "7.15", [ 7, 15, 0 ], 1126, 104, + "homology of a lie algebra with coefficients in a module", + "X83F9A1A184FB3475" ], + [ + "\033[1X\033[33X\033[0;-2YCohomology rings and Steenrod operations for fini\ +te groups\033[133X\033[101X", "8", [ 8, 0, 0 ], 1, 106, + "cohomology rings and steenrod operations for finite groups", + "X7EA6128E8703A13E" ], + [ + "\033[1X\033[33X\033[0;-2YMod-\033[22Xp\033[122X\033[101X\027\033[1X\027 co\ +homology rings of finite groups\033[133X\033[101X", "8.1", [ 8, 1, 0 ], 4, + 106, "mod-p cohomology rings of finite groups", "X877CAF8B7E64DE04" ], + [ "\033[1X\033[33X\033[0;-2YRing presentations (for the commutative \033[22X\ +p=2\033[122X\033[101X\027\033[1X\027 case)\033[133X\033[101X", "8.1-1", + [ 8, 1, 1 ], 44, 107, "ring presentations for the commutative p=2 case", + "X870E0299782638AF" ], + [ + "\033[1X\033[33X\033[0;-2YFunctorial ring homomorphisms in Mod-\033[22Xp\\ +033[122X\033[101X\027\033[1X\027 cohomology\033[133X\033[101X", "8.2", + [ 8, 2, 0 ], 72, 107, + 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"congruence subgroups cuspidal cohomology and hecke operators", "X86D5DB887ACB1661" ], [ "\033[1X\033[33X\033[0;-2YEichler-Shimura isomorphism\033[133X\033[101X", "13.1", - [ 13, 1, 0 ], 12, 139, "eichler-shimura isomorphism", "X79A1974B7B4987DE" ], - [ "\033[1X\033[33X\033[0;-2YGenerators for \033[22XSL_2( Z)\033[122X\033[101X\027\033[1X\027 and the cubic tree\033[133X\033[101X", "13.2", [ 13, 2, 0 ], 87, 140, - "generators for sl_2 z and the cubic tree", "X7BFA2C91868255D9" ], [ "\033[1X\033[33X\033[0;-2YOne-dimensional fundamental domains and generators for congruence subgroups\033[133X\033[101X", - "13.3", [ 13, 3, 0 ], 128, 141, "one-dimensional fundamental domains and generators for congruence subgroups", "X7D1A56967A073A8B" ], - [ "\033[1X\033[33X\033[0;-2YCohomology of congruence subgroups\033[133X\033[101X", "13.4", [ 13, 4, 0 ], 231, 142, "cohomology of congruence subgroups", "X818BFA9A826C0DB3" ], - [ "\033[1X\033[33X\033[0;-2YCohomology with rational 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"resolutions for nilpotent groups", + "X7B332CBE85120B38" ], + [ + "\033[1X\033[33X\033[0;-2YResolutions for groups with subnormal series\033[\ +133X\033[101X", "11.7", [ 11, 7, 0 ], 290, 126, + "resolutions for groups with subnormal series", "X7B03997084E00509" ], + [ "\033[1X\033[33X\033[0;-2YResolutions for groups with normal series\033[13\ +3X\033[101X", "11.8", [ 11, 8, 0 ], 309, 126, + "resolutions for groups with normal series", "X814FFCE080B3A826" ], + [ + "\033[1X\033[33X\033[0;-2YResolutions for polycyclic (almost) crystallograp\ +hic groups\033[133X\033[101X", "11.9", [ 11, 9, 0 ], 330, 126, + "resolutions for polycyclic almost crystallographic groups", + "X81227BF185C417AF" ], + [ + "\033[1X\033[33X\033[0;-2YResolutions for Bieberbach groups\033[133X\033[10\ +1X", "11.10", [ 11, 10, 0 ], 370, 127, "resolutions for bieberbach groups", + "X814BCDD6837BB9C5" ], + [ + "\033[1X\033[33X\033[0;-2YResolutions for arbitrary crystallographic groups\ +\033[133X\033[101X", "11.11", [ 11, 11, 0 ], 445, 128, + "resolutions for arbitrary crystallographic groups", + "X87ADCB7D7FC0B4D3" ], + [ + "\033[1X\033[33X\033[0;-2YResolutions for crystallographic groups admitting\ + cubical fundamental domain\033[133X\033[101X", "11.12", [ 11, 12, 0 ], 464, + 128, + "resolutions for crystallographic groups admitting cubical fundamental d\ +omain", "X7B9B3AF487338A9B" ], + [ + "\033[1X\033[33X\033[0;-2YResolutions for Coxeter groups\033[133X\033[101X" + , "11.13", [ 11, 13, 0 ], 499, 129, "resolutions for coxeter groups", + "X78DD8D068349065A" ], + [ "\033[1X\033[33X\033[0;-2YResolutions for Artin groups\033[133X\033[101X", + "11.14", [ 11, 14, 0 ], 525, 129, "resolutions for artin groups", + "X7C69E7227F919CC9" ], + [ + "\033[1X\033[33X\033[0;-2YResolutions for \033[22XG=SL_2( Z[1/m])\033[122X\\ +033[101X\027\033[1X\027\033[133X\033[101X", "11.15", [ 11, 15, 0 ], 543, 130, + "resolutions for g=sl_2 z[1/m]", "X8032647F8734F4EB" ], + [ + "\033[1X\033[33X\033[0;-2YResolutions for selected groups \033[22XG=SL_2( m\ +athcal O( Q(sqrtd) )\033[122X\033[101X\027\033[1X\027\033[133X\033[101X", + "11.16", [ 11, 16, 0 ], 558, 130, + "resolutions for selected groups g=sl_2 mathcal o q sqrtd", + "X7BE4DE82801CD38E" ], + [ + "\033[1X\033[33X\033[0;-2YResolutions for selected groups \033[22XG=PSL_2( \ +mathcal O( Q(sqrtd) )\033[122X\033[101X\027\033[1X\027\033[133X\033[101X", + "11.17", [ 11, 17, 0 ], 577, 130, + "resolutions for selected groups g=psl_2 mathcal o q sqrtd", + "X7D9CCB2C7DAA2310" ], + [ + "\033[1X\033[33X\033[0;-2YResolutions for a few higher-dimensional arithmet\ +ic groups\033[133X\033[101X", "11.18", [ 11, 18, 0 ], 596, 131, + "resolutions for a few higher-dimensional arithmetic groups", + "X7F699587845E6DB1" ], + [ + "\033[1X\033[33X\033[0;-2YResolutions for finite-index subgroups\033[133X\\ +033[101X", "11.19", [ 11, 19, 0 ], 618, 131, + "resolutions for finite-index subgroups", "X7812EB3F7AC45F87" ], + [ "\033[1X\033[33X\033[0;-2YSimplifying resolutions\033[133X\033[101X", + "11.20", [ 11, 20, 0 ], 645, 132, "simplifying resolutions", + "X84CAAA697FAC8E0D" ], + [ + "\033[1X\033[33X\033[0;-2YResolutions for graphs of groups and for groups w\ +ith aspherical presentations\033[133X\033[101X", "11.21", [ 11, 21, 0 ], 668, + 132, + "resolutions for graphs of groups and for groups with aspherical present\ +ations", "X780C3F038148A1C7" ], + [ + "\033[1X\033[33X\033[0;-2YResolutions for \033[22XFG\033[122X\033[101X\027\\ +033[1X\027-modules\033[133X\033[101X", "11.22", [ 11, 22, 0 ], 716, 133, + "resolutions for fg-modules", "X85AB973F8566690A" ], + [ "\033[1X\033[33X\033[0;-2YSimplicial groups\033[133X\033[101X", "12", + [ 12, 0, 0 ], 1, 134, "simplicial groups", "X7D818E5F80F4CF63" ], + [ "\033[1X\033[33X\033[0;-2YCrossed modules\033[133X\033[101X", "12.1", + [ 12, 1, 0 ], 4, 134, "crossed modules", "X808C6B357F8BADC1" ], + [ + "\033[1X\033[33X\033[0;-2YEilenberg-MacLane spaces as simplicial groups (no\ +t recommended)\033[133X\033[101X", "12.2", [ 12, 2, 0 ], 76, 135, + "eilenberg-maclane spaces as simplicial groups not recommended", + "X795E339978B42775" ], + [ + "\033[1X\033[33X\033[0;-2YEilenberg-MacLane spaces as simplicial free abeli\ +an groups (recommended)\033[133X\033[101X", "12.3", [ 12, 3, 0 ], 100, 135, + "eilenberg-maclane spaces as simplicial free abelian groups recommended" + , "X7D91E64D7DD7F10F" ], + [ + "\033[1X\033[33X\033[0;-2YElementary theoretical information on \033[22XH^\\ +342\210\227(K(\317\200,n), Z)\033[122X\033[101X\027\033[1X\027\033[133X\033[10\ +1X", "12.4", [ 12, 4, 0 ], 178, 137, + "elementary theoretical information on h^a\210\227 k i\200 n z", + "X84ABCA497C577132" ], + [ + "\033[1X\033[33X\033[0;-2YThe first three non-trivial homotopy groups of sp\ +heres\033[133X\033[101X", "12.5", [ 12, 5, 0 ], 252, 138, + "the first three non-trivial homotopy groups of spheres", + "X7F828D8D8463CC20" ], + [ + "\033[1X\033[33X\033[0;-2YThe first two non-trivial homotopy groups of the \ +suspension and double suspension of a \033[22XK(G,1)\033[122X\033[101X\027\033\ +[1X\027\033[133X\033[101X", "12.6", [ 12, 6, 0 ], 319, 139, + "the first two non-trivial homotopy groups of the suspension and double \ +suspension of a k g 1", "X81E2F80384ADF8C2" ], + [ + "\033[1X\033[33X\033[0;-2YPostnikov towers and \033[22X\317\200_5(S^3)\033[\ +122X\033[101X\027\033[1X\027\033[133X\033[101X", "12.7", [ 12, 7, 0 ], 372, + 139, "postnikov towers and i\200_5 s^3", "X83EAC40A8324571F" ], + [ + "\033[1X\033[33X\033[0;-2YTowards \033[22X\317\200_4(\316\243 K(G,1))\033[1\ +22X\033[101X\027\033[1X\027\033[133X\033[101X", "12.8", [ 12, 8, 0 ], 471, + 141, "towards i\200_4 i\244 k g 1", "X8227000D83B9A17F" ], + [ "\033[1X\033[33X\033[0;-2YEnumerating homotopy 2-types\033[133X\033[101X", + "12.9", [ 12, 9, 0 ], 532, 142, "enumerating homotopy 2-types", + "X7F5E6C067B2AE17A" ], + [ + "\033[1X\033[33X\033[0;-2YIdentifying cat\033[22X^1\033[122X\033[101X\027\\ +033[1X\027-groups of low order\033[133X\033[101X", "12.10", [ 12, 10, 0 ], + 623, 143, "identifying cat^1-groups of low order", "X7D99B7AA780D8209" ] + , + [ + "\033[1X\033[33X\033[0;-2YIdentifying crossed modules of low order\033[133X\ +\033[101X", "12.11", [ 12, 11, 0 ], 684, 144, + "identifying crossed modules of low order", "X7F386CF078CB9A20" ], + [ + "\033[1X\033[33X\033[0;-2YCongruence Subgroups, Cuspidal Cohomology and Hec\ +ke Operators\033[133X\033[101X", "13", [ 13, 0, 0 ], 1, 146, + "congruence subgroups cuspidal cohomology and hecke operators", + "X86D5DB887ACB1661" ], + [ "\033[1X\033[33X\033[0;-2YEichler-Shimura isomorphism\033[133X\033[101X", + "13.1", [ 13, 1, 0 ], 12, 146, "eichler-shimura isomorphism", + "X79A1974B7B4987DE" ], + [ + "\033[1X\033[33X\033[0;-2YGenerators for \033[22XSL_2( Z)\033[122X\033[101X\ +\027\033[1X\027 and the cubic tree\033[133X\033[101X", "13.2", [ 13, 2, 0 ], + 87, 147, "generators for sl_2 z and the cubic tree", + "X7BFA2C91868255D9" ], + [ + "\033[1X\033[33X\033[0;-2YOne-dimensional fundamental domains and generator\ +s for congruence subgroups\033[133X\033[101X", "13.3", [ 13, 3, 0 ], 128, + 148, + "one-dimensional fundamental domains and generators for congruence subgr\ +oups", "X7D1A56967A073A8B" ], + [ + "\033[1X\033[33X\033[0;-2YCohomology of congruence subgroups\033[133X\033[1\ +01X", "13.4", [ 13, 4, 0 ], 231, 149, "cohomology of congruence subgroups", + "X818BFA9A826C0DB3" ], + [ + "\033[1X\033[33X\033[0;-2YCohomology with rational coefficients\033[133X\\ +033[101X", "13.4-1", [ 13, 4, 1 ], 327, 151, + "cohomology with rational coefficients", "X7F55F8EA82FE9122" ], + [ "\033[1X\033[33X\033[0;-2YCuspidal cohomology\033[133X\033[101X", "13.5", + [ 13, 5, 0 ], 361, 151, "cuspidal cohomology", "X84D30F1580CD42D1" ], + [ + "\033[1X\033[33X\033[0;-2YHecke operators on forms of weight 2\033[133X\\ +033[101X", "13.6", [ 13, 6, 0 ], 464, 153, + "hecke operators on forms of weight 2", "X80861D3F87C29C43" ], + [ + "\033[1X\033[33X\033[0;-2YHecke operators on forms of weight \033[22X\342\\ +211\245 2\033[122X\033[101X\027\033[1X\027\033[133X\033[101X", "13.7", + [ 13, 7, 0 ], 537, 154, "hecke operators on forms of weight a\211\246 2" + , "X831BB0897B988DA3" ], + [ + "\033[1X\033[33X\033[0;-2YReconstructing modular forms from cohomology comp\ +utations\033[133X\033[101X", "13.8", [ 13, 8, 0 ], 556, 154, + "reconstructing modular forms from cohomology computations", + "X84CC51EE8525E0D9" ], + [ "\033[1X\033[33X\033[0;-2YThe Picard group\033[133X\033[101X", "13.9", + [ 13, 9, 0 ], 675, 156, "the picard group", "X8180E53C834301EF" ], + [ "\033[1X\033[33X\033[0;-2YBianchi groups\033[133X\033[101X", "13.10", + [ 13, 10, 0 ], 816, 158, "bianchi groups", "X858B1B5D8506FE81" ], + [ + "\033[1X\033[33X\033[0;-2YSome other infinite matrix groups\033[133X\033[10\ +1X", "13.11", [ 13, 11, 0 ], 956, 159, "some other infinite matrix groups", + "X86A6858884B9C05B" ], + [ + "\033[1X\033[33X\033[0;-2YIdeals and finite quotient groups\033[133X\033[10\ +1X", "13.12", [ 13, 12, 0 ], 1068, 161, "ideals and finite quotient groups", + "X7EF5D97281EB66DA" ], + [ + "\033[1X\033[33X\033[0;-2YCongruence subgroups for ideals\033[133X\033[101X\ +", "13.13", [ 13, 13, 0 ], 1180, 163, "congruence subgroups for ideals", + "X7D1F72287F14C5E1" ], + [ "\033[1X\033[33X\033[0;-2YFirst homology\033[133X\033[101X", "13.14", + [ 13, 14, 0 ], 1252, 164, "first homology", "X85E912617AFE03F4" ], + [ "\033[1X\033[33X\033[0;-2YParallel computation\033[133X\033[101X", "14", + [ 14, 0, 0 ], 1, 166, "parallel computation", "X7F571E8F7BBC7514" ], + [ + "\033[1X\033[33X\033[0;-2YAn embarassingly parallel computation\033[133X\\ +033[101X", "14.1", [ 14, 1, 0 ], 4, 166, + "an embarassingly parallel computation", "X7EAE286B837D27BA" ], + [ + "\033[1X\033[33X\033[0;-2YAn non-embarassingly parallel computation\033[133\ +X\033[101X", "14.2", [ 14, 2, 0 ], 35, 166, + "an non-embarassingly parallel computation", "X7AA9C5B27A8322D0" ], + [ + "\033[1X\033[33X\033[0;-2YRegular CW-structure on knots (written by Kelvin \ +Killeen)\033[133X\033[101X", "15", [ 15, 0, 0 ], 1, 168, + "regular cw-structure on knots written by kelvin killeen", + "X7C57D4AB8232983E" ], + [ + "\033[1X\033[33X\033[0;-2YKnot complements in the 3-ball\033[133X\033[101X" + , "15.1", [ 15, 1, 0 ], 4, 168, "knot complements in the 3-ball", + "X86F56A85848347FF" ], + [ "\033[1X\033[33X\033[0;-2YTubular neighbourhoods\033[133X\033[101X", + "15.2", [ 15, 2, 0 ], 93, 169, "tubular neighbourhoods", + "X83EA2A38801E7A4C" ], + [ + "\033[1X\033[33X\033[0;-2YKnotted surface complements in the 4-ball\033[133\ +X\033[101X", "15.3", [ 15, 3, 0 ], 265, 172, + "knotted surface complements in the 4-ball", "X78C28038837300BD" ], + [ "Bibliography", "bib", [ "Bib", 0, 0 ], 1, 179, "bibliography", + "X7A6F98FD85F02BFE" ], + [ "References", "bib", [ "Bib", 0, 0 ], 1, 179, "references", + "X7A6F98FD85F02BFE" ], + [ "Index", "ind", [ "Ind", 0, 0 ], 1, 182, "index", "X83A0356F839C696F" ] ] ); diff --git a/tutorial/mybib.xml b/tutorial/mybib.xml index 58a1212a..e6a60f9d 100644 --- a/tutorial/mybib.xml +++ b/tutorial/mybib.xml @@ -53,6 +53,40 @@ 2015 + + + + DavidCoeurjolly +BertrandKerautret + Jacques-OlivierLachaud + + +Extraction of Connected Region Boundary in +Multidimensional Images + + Image Processing On Line + + 2014 + + + + + +D.Martin +C.Fowlkes +D.Tal +J.Malik + + + A Database of Human Segmented Natural Images and its + Application to Evaluating Segmentation Algorithms and + Measuring Ecological Statistics + +Proc. 8th Int'l Conf. Computer Vision, 2, pp 416--423 + +2001 + + diff --git a/tutorial/mybib.xml.bib b/tutorial/mybib.xml.bib index 6b8ae2b0..5fde229f 100644 --- a/tutorial/mybib.xml.bib +++ b/tutorial/mybib.xml.bib @@ -25,11 +25,29 @@ @book{ thurston } @book{ goncalves, author = {Goncalves, D. and Martins, S.}, - title = {Diagonal approximation and the cohomology ring of the fundamental groups of surfaces}, + title = {Diagonal approximation and the cohomology ring of the + fundamental groups of surfaces}, publisher = {European Journal of Mathematics, 1, pp122--137}, year = {2015}, printedkey = {GM15} } +@book{ coeurjolly, + author = {Coeurjolly, D. and Kerautret, B. and Lachaud, J.-O.}, + title = {Extraction of Connected Region Boundary in + Multidimensional Images}, + publisher = {Image Processing On Line}, + year = {2014}, + printedkey = {CKL14} +} +@book{ MartinFTM01, + author = {Martin, D. and Fowlkes, C. and Tal, D. and Malik, J.}, + title = {A Database of Human Segmented Natural Images and its + Application to Evaluating Segmentation Algorithms and + Measuring Ecological Statistics}, + publisher = {Proc. 8th Int'l Conf. Computer Vision, 2, pp 416--423}, + year = {2001}, + printedkey = {MFTM01} +} @book{ johnson, author = {Johnson, F.}, title = {Syzygies and dihedral resolutions for dihedral groups}, @@ -67,14 +85,16 @@ @book{ lmoser } @book{ horadam, author = {Horadam, K.}, - title = {An introduction to cocyclic generalised Hadamard matrices}, + title = {An introduction to cocyclic generalised Hadamard + matrices}, publisher = {Discrete Applied Math, 102, 115-130}, year = {2000}, printedkey = {Hor00} } @book{ dpr, author = {Dijkgraaf, R. and Pasquier, V. and Roche, P.}, - title = {Quasi-Hopf algebras, group cohomology and orbifold models}, + title = {Quasi-Hopf algebras, group cohomology and orbifold + models}, publisher = {Nuclear Phys. B Proc. Suppl. 18B, 60-72}, year = {1991}, printedkey = {DPR91} @@ -95,7 +115,8 @@ @book{ reidemeister } @book{ moise, author = {Moise, E.}, - title = {Affine structures in 3-manifolds V. The triangulation theorem and Hauptvermu- tung}, + title = {Affine structures in 3-manifolds V. The triangulation + theorem and Hauptvermu- tung}, publisher = {Annals of Math. 56, 96--114}, year = {1952}, printedkey = {Moi52} @@ -109,14 +130,16 @@ @book{ przytycki } @book{ tomoda, author = {Tomoda, S. and Zvengrowski, P.}, - title = {Remarks on the cohomology of finite fundamental groups of 3-manifolds}, + title = {Remarks on the cohomology of finite fundamental groups + of 3-manifolds}, publisher = {Geometry and Topology Monographs 14, 519-556}, year = {2008}, printedkey = {TZ08} } @book{ spreerkhuenel, author = {Spreer, J. and Khuenel, W.}, - title = {Combinatorial properties of the K3 surface: Simplicial blowups and slicings}, + title = {Combinatorial properties of the K3 surface: Simplicial + blowups and slicings}, publisher = {Experimental Mathematics Volume 20 Issue 2}, year = {2011}, printedkey = {SK11} @@ -124,7 +147,9 @@ @book{ spreerkhuenel @book{ milnor, author = {Milnor, J.}, title = {On simply connected 4-manifolds}, - publisher = {International symposium on algebraic topology, Universidad Nacional Autonoma de Mexico and UNESCO, Mexico City}, + publisher = {International symposium on algebraic topology, + Universidad Nacional Autonoma de Mexico and UNESCO, + Mexico City}, year = {1958}, printedkey = {Mil58} } @@ -138,7 +163,8 @@ @book{ hatcher } @book{ ksontini, author = {Ksontini, R.}, - title = {Proprietes homotopiques du complexe de Quillen du groupe symetrique}, + title = {Proprietes homotopiques du complexe de Quillen du + groupe symetrique}, publisher = {These de doctorat, Universitet de Lausanne}, year = {2000}, url = {http://sma.epfl.ch/\texttt{\symbol{126}}thevenaz/theses/Ksontini{\textunderscore}thesis.pdf}, @@ -147,15 +173,18 @@ @book{ ksontini @book{ bergeron, author = {Bergeron, N.}, title = {Torsion homology growth in arithmetic groups}, - publisher = {EuropeanMathematical Society, European Congress of Mathematicians, July 18-22}, + publisher = {EuropeanMathematical Society, European Congress of + Mathematicians, July 18-22}, year = {2016}, url = {https://www.ems-ph.org/books/book.php?proj{\textunderscore}nr=220}, printedkey = {Ber16} } @book{ rahmthesis, author = {Rahm, A.}, - title = {Cohomologies and $K$-theory of Bianchi groups using computational geometric models}, - publisher = {These de doctorat, Universite Joseph-Fourier -- Grenoble I}, + title = {Cohomologies and $K$-theory of Bianchi groups using + computational geometric models}, + publisher = {These de doctorat, Universite Joseph-Fourier -- + Grenoble I}, year = {2010}, url = {https://tel.archives-ouvertes.fr/tel-00526976v6}, printedkey = {Rah10} @@ -199,8 +228,10 @@ @article{ sengun printedkey = {Sen11} } @article{ schoennenbeck, - author = {Braun, O. and Coulangeon, R. and Nebe, G. and Schoennenbeck, S.}, - title = {Computing in arithmetic groups with Vorono{\"\i}{\textquoteright}s algorithm}, + author = {Braun, O. and Coulangeon, R. and Nebe, G. and + Schoennenbeck, S.}, + title = {Computing in arithmetic groups with + Vorono{\"\i}{\textquoteright}s algorithm}, journal = {J. Algebra}, volume = {435}, year = {2015}, @@ -209,7 +240,8 @@ @article{ schoennenbeck } @article{ swan, author = {Swan, R.}, - title = {Generators and relations for certain general linear groups}, + title = {Generators and relations for certain general linear + groups}, journal = {Advances in Mathematics}, volume = {6}, year = {1971}, @@ -228,7 +260,8 @@ @article{ atkinlehner } @article{ kulkarni, author = {Kulkarni, R.}, - title = {An arithmetic-geometric method in the study of the subgroups of the modular group}, + title = {An arithmetic-geometric method in the study of the + subgroups of the modular group}, journal = {American Journal of Mathematics}, volume = {113, No. 6}, year = {1991}, @@ -266,7 +299,8 @@ @article{ MR1758871 } @article{ MR2441256, author = {Kauffman, L. H. and Faria Martins, J.}, - title = {Invariants of welded virtual knots via crossed module invariants of knotted surfaces}, + title = {Invariants of welded virtual knots via crossed module + invariants of knotted surfaces}, journal = {Compos. Math.}, volume = {144}, number = {4}, diff --git a/tutorial/tutex/14.2a.txt b/tutorial/tutex/14.2a.txt index ee324020..294c406b 100644 --- a/tutorial/tutex/14.2a.txt +++ b/tutorial/tutex/14.2a.txt @@ -1,3 +1,4 @@ +gap> F:=FreeGroup(2);;D:=F/[F.1^2,F.1*F.2*F.1^-1*F.2^-2];; gap> R:=ResolutionSmallGroup(D,15);; gap> Size(R); [ 4, 7, 8, 6, 4, 8, 8, 6, 4, 8, 8, 6, 4, 8, 8 ] diff --git a/tutorial/tutex/14.2c.txt b/tutorial/tutex/14.2c.txt new file mode 100644 index 00000000..47799da8 --- /dev/null +++ b/tutorial/tutex/14.2c.txt @@ -0,0 +1,16 @@ +gap> Y:=PoincareDodecahedronCWComplex( +> [[1,2,3,4,5],[6,7,8,9,10]], +> [[1,11,16,12,2],[19,9,8,18,14]], +> [[2,12,17,13,3],[20,10,9,19,15]], +> [[3,13,18,14,4],[16,6,10,20,11]], +> [[4,14,19,15,5],[17,7,6,16,12]], +> [[5,15,20,11,1],[18,8,7,17,13]]);; +gap> G:=FundamentalGroup(Y); +<fp group on the generators [ f1, f2 ]> +gap> RelatorsOfFpGroup(G); +[ f2^-1*f1^-1*f2*f1^-1*f2^-1*f1, f2^-1*f1*f2^2*f1*f2^-1*f1^-1 ] +gap> StructureDescription(G); +"SL(2,5)" +gap> R:=ResolutionSmallGroup(G,3);; +gap> List([0..3],R!.dimension); +[ 1, 2, 2, 1 ] diff --git a/tutorial/tutex/4.17.txt b/tutorial/tutex/4.17.txt new file mode 100644 index 00000000..1e45bc54 --- /dev/null +++ b/tutorial/tutex/4.17.txt @@ -0,0 +1,11 @@ +gap> file:=Filename(DirectoriesPackageLibrary("HAP"),"../tutorial/images/circularGradient.png");; +gap> L:=[];; +gap> for n in [1..15] do +> M:=ReadImageAsPureCubicalComplex(file,n*30000); +> M:=PureComplexBoundary(M);; +> Add(L,M); +> od;; +gap> C:=L[1];; +gap> for n in [2..Length(L)] do C:=PureComplexUnion(C,L[n]); od; +gap> Display(C); +gap> Display(ContractedComplex(C)); diff --git a/tutorial/tutex/4.18.txt b/tutorial/tutex/4.18.txt new file mode 100644 index 00000000..d8b8a6ab --- /dev/null +++ b/tutorial/tutex/4.18.txt @@ -0,0 +1,3 @@ +gap> F:=ReadImageAsFilteredPureCubicalComplex(file,20);; +gap> P:=PersistentBettiNumbersAlt(F,1);; +gap> BarCodeCompactDisplay(P); diff --git a/tutorial/tutex/4.19.txt b/tutorial/tutex/4.19.txt new file mode 100644 index 00000000..6d9ec710 --- /dev/null +++ b/tutorial/tutex/4.19.txt @@ -0,0 +1,18 @@ +gap> file:=Filename(DirectoriesPackageLibrary("HAP"),"../tutorial/images/circularGradient.png");; +gap> F:=ReadImageAsFilteredPureCubicalComplex(file,20);; +gap> FF:=ComplementOfFilteredPureCubicalComplex(F); + +gap> W:=(FiltrationTerm(FF,3)); +gap> for n in [4..23] do +> L:=[];; +> for i in [1..PathComponentOfPureComplex(W,0)] do +> P:=PathComponentOfPureComplex(W,i);; +> Q:=ThickeningFiltration(P,150,FiltrationTerm(FF,n));; +> Add(L,Q);; +> od;; +> W:=Basins(L); +> od; + +gap> C:=PureComplexComplement(W);; +gap> T:=PureComplexThickened(C);; C:=ContractedComplex(T,C);; +gap> Display(C); diff --git a/tutorial/tutex/4.5.txt b/tutorial/tutex/4.5.txt index 35f76c03..6955ea27 100644 --- a/tutorial/tutex/4.5.txt +++ b/tutorial/tutex/4.5.txt @@ -1,5 +1,5 @@ gap> file:=HapFile("image1.3.2.png");; -gap> F:=ReadImageAsFilteredPureCubicalComplex(file,20); +gap> F:=ReadImageAsFilteredPureCubicalComplex(file,40); Filtered pure cubical complex of dimension 2. gap> P:=PersistentBettiNumbers(F,0);; gap> BarCodeCompactDisplay(P); diff --git a/tutorial/tutex/4.6a.txt b/tutorial/tutex/4.6a.txt index e3513764..23782c97 100644 --- a/tutorial/tutex/4.6a.txt +++ b/tutorial/tutex/4.6a.txt @@ -1,8 +1,8 @@ -gap> Y:=FiltrationTerm(F,4); +gap> F:=ReadImageAsFilteredPureCubicalComplex(file,500);; +gap> Y:=FiltrationTerm(F,64); Pure cubical complex of dimension 2. gap> BettiNumber(Y,0); 20 gap> BettiNumber(Y,1); 14 gap> Display(Y); - diff --git a/tutorial/tutex/4.6b.txt b/tutorial/tutex/4.6b.txt index 24f31050..4dcda845 100644 --- a/tutorial/tutex/4.6b.txt +++ b/tutorial/tutex/4.6b.txt @@ -1,5 +1,5 @@ -gap> F:=ReadImageAsFilteredPureCubicalComplex("my_coins.png",30);; -gap> M:=FiltrationTerm(F,21);; #Chosen after viewing degree 0 barcode for F +gap> F:=ReadImageAsFilteredPureCubicalComplex("my_coins.png",40);; +gap> M:=FiltrationTerm(F,24);; #Chosen after viewing degree 0 barcode for F gap> M:=PureComplexThickened(M);; gap> M:=PureComplexThickened(M);; gap> C:=PureComplexComplement(M);; diff --git a/tutorial/tutex/4.6d.txt b/tutorial/tutex/4.6d.txt new file mode 100644 index 00000000..14ef04de --- /dev/null +++ b/tutorial/tutex/4.6d.txt @@ -0,0 +1,7 @@ +gap> F:=ReadImageAsFilteredPureCubicalComplex(file,500);; +gap> L:=[20,60,61,62,63,64,65,66,67,68,69,70];; +gap> T:=FiltrationTerms(F,L);; +gap> P0:=PersistentBettiNumbers(T,0);; +gap> BarCodeCompactDisplay(P0); +gap> P1:=PersistentBettiNumbers(T,1);; +gap> BarCodeCompactDisplay(P1); diff --git a/tutorial/tutex/4.6g.txt b/tutorial/tutex/4.6g.txt new file mode 100644 index 00000000..7ad526b6 --- /dev/null +++ b/tutorial/tutex/4.6g.txt @@ -0,0 +1,27 @@ +gap> file:=Filename(DirectoriesPackageLibrary("HAP"),"../tutorial/images/3096b.jpg");; +gap> F:=ReadImageAsFilteredPureCubicalComplex(file,30);; +gap> F:=ComplementOfFilteredPureCubicalComplex(F);; +gap> M:=FiltrationTerm(F,27);; #Thickening chosen based on degree 0 barcode +gap> Display(M);; +gap> P:=List([1..BettiNumber(M,0)],n->PathComponentOfPureComplex(M,n));; +gap> P:=Filtered(P,m->Size(m)>10);; +gap> M:=P[1];; +gap> for m in P do +> M:=PureComplexUnion(M,m);; +> od; +gap> T:=ThickeningFiltration(M,50);; +gap> BettiNumber(FiltrationTerm(T,11),0); +1 +gap> BettiNumber(FiltrationTerm(T,11),1); +1 +gap> BettiNumber(FiltrationTerm(T,12),1); +0 +gap> #Confirmation that 11-th filtration term has one hole and the 12-th term is contractible. +gap> C:=FiltrationTerm(T,11);; +gap> for n in Reversed([1..10]) do +> C:=ContractedComplex(C,FiltrationTerm(T,n)); +> od; +gap> C:=PureComplexBoundary(PureComplexThickened(C));; +gap> H:=HomotopyEquivalentMinimalPureCubicalSubcomplex(FiltrationTerm(T,12),C);; +gap> B:=ContractedComplex(PureComplexBoundary(H));; +gap> Display(B); diff --git a/tutorial/tutex/4.7c.txt b/tutorial/tutex/4.7c.txt new file mode 100644 index 00000000..b0ca1a70 --- /dev/null +++ b/tutorial/tutex/4.7c.txt @@ -0,0 +1,16 @@ +gap> Y2:=FiltrationTerm(F,10);; +gap> for t in Reversed([1..9]) do +> Y2:=ContractedComplex(Y2,FiltrationTerm(F,t)); +> od; +gap> Y2:=ContractedComplex(Y2);; + +gap> Size(FiltrationTerm(F,10)); +918881 +gap> Size(Y2); +61618 + +gap> Y1:=PureComplexDifference(Y2,PureComplexRandomCell(Y2));; +gap> Y1:=ContractedComplex(Y1);; +gap> Size(Y1); +474 +gap> Display(Y1); diff --git a/tutorial/tutex/5.13.txt b/tutorial/tutex/5.13.txt new file mode 100644 index 00000000..407b695c --- /dev/null +++ b/tutorial/tutex/5.13.txt @@ -0,0 +1,9 @@ +gap> n:=3;;c:=1;; +gap> G:=Image(NqEpimorphismNilpotentQuotient(FreeGroup(n),c));; +gap> R:=ResolutionNilpotentGroup(G,4);; +gap> P:=PresentationOfResolution(R);; +gap> P.freeGroup; +<free group on the generators [ x, y, z ]> +gap> P.relators; +[ y^-1*x^-1*y*x, z^-1*x^-1*z*x, z^-1*y^-1*z*y ] +gap> IdentityAmongRelatorsDisplay(R,1); diff --git a/tutorial/tutex/5.14.txt b/tutorial/tutex/5.14.txt new file mode 100644 index 00000000..008019f9 --- /dev/null +++ b/tutorial/tutex/5.14.txt @@ -0,0 +1,9 @@ +gap> n:=2;;c:=2;; +gap> G:=Image(NqEpimorphismNilpotentQuotient(FreeGroup(n),c));; +gap> R:=ResolutionNilpotentGroup(G,4);; +gap> P:=PresentationOfResolution(R);; +gap> P.freeGroup; +<free group on the generators [ x, y, z ]> +gap> P.relators; +[ z*x*y*x^-1*y^-1, z*x*z^-1*x^-1, z*y*z^-1*y^-1 ] +gap> IdentityAmongRelatorsDisplay(R,1); diff --git a/tutorial/tutex/6.31.txt b/tutorial/tutex/6.31.txt new file mode 100644 index 00000000..c9d8a281 --- /dev/null +++ b/tutorial/tutex/6.31.txt @@ -0,0 +1,9 @@ +gap> M:=FullMatrixAlgebra(Rationals,4);; +gap> A:=LieAlgebra(M);; +gap> V:=AdjointModule(A);; +gap> C:=ChevalleyEilenbergComplex(V,17);; +gap> List([0..17],C!.dimension); +[ 16, 256, 1920, 8960, 29120, 69888, 128128, 183040, 205920, 183040, 128128, + 69888, 29120, 8960, 1920, 256, 16, 0 ] +gap> Homology(C,4); +1 diff --git a/tutorial/tutorialGroupCohomology.xml b/tutorial/tutorialGroupCohomology.xml index fe112af7..51332dc7 100644 --- a/tutorial/tutorialGroupCohomology.xml +++ b/tutorial/tutorialGroupCohomology.xml @@ -1,4 +1,4 @@ -Cohomology of groups +Cohomology of groups (and Lie Algebras)
    Finite groups Naive homology computation for a very small group @@ -612,5 +612,17 @@ that the Poincare series <#Include SYSTEM "tutex/6.28.txt"> +
    + +
    Homology of a Lie algebra with coefficients in a module + + Let A be the Lie algebra constructed from the associative algebra M^{4\times 4}(\mathbb Q) of all 4\times 4 rational matrices. Let V be its adjoint module (with underlying vector space of dimension 16 and + equal to that of A). The following commands compute H_{4}(A,V) = \mathbb Q. + + +<#Include SYSTEM "tutex/6.31.txt"> + + +

    Note that the eighth term C_{8}(V) in the Chevalley-Eilenberg complex C_\ast(V) is a vector space of dimension 205920 and so it will take longer to compute the homology in degree 8.

    diff --git a/tutorial/tutorialGroupTheoretic.xml b/tutorial/tutorialGroupTheoretic.xml index 2ffe5811..3492dcbc 100644 --- a/tutorial/tutorialGroupTheoretic.xml +++ b/tutorial/tutorialGroupTheoretic.xml @@ -53,6 +53,65 @@ establish that the 2-complex associated to the group presentation < <#Include SYSTEM "tutex/5.3.txt"> + +
    Group presentations and homotopical syzygies + Free resolutons for a group G are constructed in HAP + as the cellular chain complex R_\ast=C_\ast(\tilde X) of the universal cover of some + CW-complex X=K(G,1). The 2-skeleton of + X gives rise to a free presentation for the group G. + This presentation depends on a choice of maximal tree in the 1-skeleton of X in cases where X has more than one 0-cell. The attaching maps of + 3-cells in X can be regarded as + homotopical syzygies or van Kampen diagrams over the group presentation whose boundaries spell the trivial word. + +

    The following example constructs four terms of + a resolution for the free abelian group G on n=3 generators, and then extracts the group presentation + from the resolution as well as the unique homotopical syzygy. The syzygy is visualized in terms of its graph of edges, + directed edges being coloured according to the corresponding + group generator. (In this example the CW-complex \tilde X is regular, but in cases where it is not the visualization may be a quotient of the 1-skeleton of the syzygy.) + +<#Include SYSTEM "tutex/5.13.txt"> + + +

    +<img src="images/syzfab.gif" align="center" height="160" alt="Homotopical syzygy for the free abelian group on three generators"/> + + +

    + This homotopical syzygy represents a relationship between the three relators [x,y], [x,z] and [y,z] where [x,y]=xyx^{-1}y^{-1}. The syzygy can be thought of as a geometric relationship between commutators corresponding to the well-known Hall-Witt identity: +

    [\ [x,y],\ {^yz}\ ]\ \ [\ [y,z],\ {^zx}\ ]\ \ [\ [z,x],\ {^xy}\ ]\ \ =\ \ 1\ \ . + +

    The homotopical syzygy is special since in this example the edge directions and labels can be understood as specifying three homeomorphisms + between pairs of faces. Viewing the syzygy as the boundary of the 3-ball, by using the homeomorphisms to identify the faces in each face pair we obtain a quotient CW-complex M + involving one vertex, three edges, three 2-cells and one 3-cell. The cell structure on the quotient exists because, + under the restrictions of homomorphisms to the edges, any cycle of edges retricts to the identity map on any given edge. The following + result tells us that M is in fact a closed oriented compact 3-manifold. + +

    Theorem. [Seifert u. Threlfall, Topologie, p.208] Let S^2 denote the boundary + of the 3-ball B^3 and suppose + that the sphere S^2 is given a regular CW-structure in which the faces are partitioned into a collection of face pairs. Suppose that for each face pair there is an orientation reversing homeomorphism between the two faces that sends edges to edges and vertices + to vertices. Suppose that by using these homeomorphisms to identity face pairs we obtain a (not necessarily regular) CW-structure on the quotient M. Then M is a closed compact orientable manifold if and only if its Euler characteristic is \chi(M)=0. + +

    The next commands construct a presentation and associated unique homotopical syzygy for the free nilpotent group of class c=2 on n=2 generators. + + +<#Include SYSTEM "tutex/5.14.txt"> + + + +

    +<img src="images/syznil.gif" align="center" height="160" alt="Homotopical syzygy for the free nilpotent group of class two on two generators"/> + + +

    The syzygy represents the following relationship between commutators (in a free group). +

    [\ [x^{-1},y][x,y]\ ,\ [y,x][y^{-1},x]y^{-1}\ ]\ [\ [y,x][y^{-1},x]\ + , \ x^{-1} \ ] \ \ =\ \ 1 + + +

    + Again, using the theorem of Seifert and Threlfall we see that the free nilpotent group of class two on two generators arises as + the fundamental group of a closed compact orientable 3-manifold + M. +

    Bogomolov multiplier diff --git a/tutorial/tutorialResolutions.xml b/tutorial/tutorialResolutions.xml index e89b35b7..0b125b61 100644 --- a/tutorial/tutorialResolutions.xml +++ b/tutorial/tutorialResolutions.xml @@ -23,11 +23,11 @@ resolution. <#Include SYSTEM "tutex/14.2.txt"> The suspicion that this resolution R_\ast -is periodic of period 4 can be verified by +is periodic of period 4 can be confirmed by constructing the chain complex C_\ast=R_\ast\otimes_{\mathbb Z}\mathbb ZG and verifying that boundary matrices repeat with period 4.

    A second example of a periodic resolution, for the Dihedral group -D_{2k+1}=\langle x, y\ |\ x^2= xy^kx^{-1}y^{-k-1}\rangle of order 2k+2 in the case k=1, is constructed and verified for periodicity in the next example. +D_{2k+1}=\langle x, y\ |\ x^2= xy^kx^{-1}y^{-k-1} = 1\rangle of order 2k+2 in the case k=1, is constructed and verified for periodicity in the next example. <#Include SYSTEM "tutex/14.2a.txt"> @@ -46,6 +46,13 @@ A slightly different periodic resolution for D_{2k+1} has been obtain mor <#Include SYSTEM "tutex/14.2b.txt"> +

    The performance of the + function ResolutionSmallGroup(G,n) is very sensistive to the choice of presentation for the input group G. If G + is an fp-group then the defining presentation for G is used. If G + is a permutaion group or finite matrix group then GAP functions are invoked to find a presentation for G. The following commands use a geometrically derived presentation for SL(2,5) as input in order to obtain the first few terms of a periodic resolution for this group of period 4. + +<#Include SYSTEM "tutex/14.2c.txt"> +

    diff --git a/tutorial/tutorialTDA.xml b/tutorial/tutorialTDA.xml index 5e2c5df7..4aad3f27 100644 --- a/tutorial/tutorialTDA.xml +++ b/tutorial/tutorialTDA.xml @@ -12,6 +12,8 @@ filtration of length 100 on the first two dimensions of the assotiated cl <img src="images/bar0.png" align="center" height="60" alt="degree 0 barcode"/> +

    The first 54 terms in the filtration each have 74 path components -- one for each point in the sample. During the next 9 filtration terms the number of path components reduces, meaning that sample points begin to coalesce due to the + formation of edges in the simplicial complexes. Then, two path components persist over an interval of 18 filtration terms, before they eventually coalesce.

    The next commands display the resulting degree 1 persistent homology as a barcode. @@ -22,7 +24,8 @@ The next commands display the resulting degree 1 persistent homology as a <img src="images/bar1.png" align="center" height="120" alt="degree 1 bar code"/> -

    The following command displays the 1 skeleton of the simplicial complex arizing as the 65-th term in the filtration on the clique complex. +

    Interpreting short bars as noise, we see for instance that + the 65th term in the filtration could be regarded as noiseless and belonging to a "stable interval" in the filtration with regards to first and second homology functors. The following command displays (up to homotopy) the 1 skeleton of the simplicial complex arizing as the 65-th term in the filtration on the clique complex. <#Include SYSTEM "tutex/4.3.txt"> @@ -30,13 +33,13 @@ The next commands display the resulting degree 1 persistent homology as a <img src="images/twocircles.png" align="center" height="300" alt="1-skeleton"/> -

    These computations suuggest that the dataset contains two persistent +

    These computations suggest that the dataset contains two persistent path components (or clusters), and that each path component is in some sense periodic. The final command displays one possible representation of the data as points on two circles. Background to the data -

    Each point in the dataset was an image consisting of 732\times 761 pixels. This point was regarded as a vector in \mathbb R^{732\times 761} and the matrix D was constructed using the Euclidean metric. The images were the following: +

    Each point in the dataset was an image consisting of 732\times 761 pixels. This point was regarded as a vector in \mathbb R^{557052}=\mathbb R^{732\times 761} and the matrix D was constructed using the Euclidean metric. The images were the following:

    <img src="images/letters.png" align="center" height="220" alt="letters"/> @@ -47,7 +50,7 @@ path components (or clusters), and that each path component is in some sense per

    Mapper clustering

    The following example reads in a set S of vectors of rational numbers. It uses the Euclidean distance d(u,v) between vectors. It fixes -some vector $u_0\in S$ and uses the associated + some vector u_0\in S and uses the associated function f\colon D\rightarrow [0,b] \subset \mathbb R, v\mapsto d(u_0,v). In addition, it uses an open cover of the interval [0,b] consisting of 100 uniformly distributed overlapping open subintervals of radius r=29. It also uses a simple clustering algorithm implemented in the function cluster. @@ -57,7 +60,7 @@ produce a simplicial complex M which is intended to be a representation of the data. The complex M is 1-dimensional and the final command uses GraphViz software to visualize the graph. The nodes of this simplicial -complex are "buckets" containing data points. A data point may reside in several buckets. The number of points in the bucket determines the size of the node. Two nodes are connected by an edge when their end-point nodes contain common data points. +complex are "buckets" containing data points. A data point may reside in several buckets. The number of points in the bucket determines the size of the node. Two nodes are connected by an edge when they contain common data points. <#Include SYSTEM "tutex/4.4.txt"> @@ -76,6 +79,57 @@ complex are "buckets" containing data points. A data point may reside in several

    +
    Some tools for handling pure complexes + A CW-complex X is said to be pure if all of its top-dimensional cells have a common dimension. There are instances where such + a space X provides a convenient ambient space whose subspaces can be used to model experimental data. For instance, the plane X=\mathbb R^2 admits a pure regular CW-structure whose 2-cells are + open unit squares with integer coordinate vertices. An alternative, and sometimes preferrable, pure regular CW-structure on \mathbb R^2 is one where the 2-cells are all reguar hexagons with sides of unit length. Any digital image can be thresholded to produce a black-white + image and this black-white image can naturally be regared as a finite pure cellular subcomplex of either of the two proposed CW-structures on \mathbb R^2. + Analogously, thresholding can be used to represent 3-dimensional greyscale images as finite pure cellular subspaces + of cubical or permutahedral CW-structures on \mathbb R^3, and to represent RGB colour photographs as + analogous subcomplexes of \mathbb R^5. + +

    In this section we list a few functions for performing basic operations on n-dimensional + pure cubical and pure permutahedral finite subcomplexes M of X=R^n. We refer to M simply as a pure complex. In subsequent sections we demonstrate how these few functions on pure complexes allow for in-depth analysis of experimental data. + +

    (Aside. The basic operations could equally well be implemented for other CW-decompositions of X=\mathbb R^n such as the regular CW-decompositions arising as the tessellations by a fundamental domain of a Bieberbach group (=torsion free crytallographic group). Moreover, the basic operations could also be implemented for + other manifolds such as an n-torus X=S^1\times S^1 \times \cdots \times S^1 or n-sphere X=S^n or for X the universal cover of some interesting hyperbolic 3-manifold. An example use of the ambient manifold X=S^1\times S^1\times S^1 could be for the construction of a cellular subspace recording the time of day, day of week and week of the year of crimes committed in a population.) + +

    Basic operations returning pure complexes. ( Function descriptions available ../doc/chap1_mj.html#X7FD50DF6782F00A0here.) + + + PureCubicalComplex(binary array) + PurePermutahedralComplex(binary array) + ReadImageAsPureCubicalComplex(file,threshold) + + ReadImageSquenceAsPureCubicalComplex(file,threshold) + + PureComplexBoundary(M) + + PureComplexComplement(M) + PureComplexRandomCell(M) + + PureComplexThickened(M) + ContractedComplex(M, optional subcomplex of M) + ExpandedComplex(M, optional supercomplex of M) + + + PureComplexUnion(M,N) + + PureComplexIntersection(M,N) + + PureComplexDifference(M,N) + + FiltrationTerm(F,n) + + +

    Basic operations returning filtered pure complexes. + + PureComplexThickeningFiltration(M,length) + + ReadImageAsFilteredPureCubicalComplex(file,length) + + +

    Digital image analysis and persistent homology

    The following example reads in a digital image as a filtered @@ -98,20 +152,39 @@ The filtration is obtained by thresholding at a sequence of uniformly spaced val

    The 20 persistent bars in the -degree 0 barcode suggest that the image has 20 objects. The degree 1 barcode suggests that 14 (or possibly 17) of these objects have holes in them. +degree 0 barcode suggest that the image has 20 objects. The degree 1 barcode suggests that there are 14 (or possibly 17) holes in these 20 objects. Naive example of image segmentation by automatic thresholding Assuming that short bars and isolated points in the barcodes represent noise while long bars represent essential features, a "noiseless" representation of the image should correspond to a term in the filtration corresponding to - a column in the barcode incident with all long bars but incident with no short bars or isolated points. The following commands confirm that the 4th term in the filtration is such a term and display this term as a binary image. + a column in the barcode incident with all the long bars but incident with no short bars or isolated points. There is no noiseless term in the above filtration of length 40. However (in conjunction with the next subsection) the following commands confirm that the 64th term in the filtration of length 500 is such a term and display this term as a binary image. <#Include SYSTEM "tutex/4.6a.txt"> + <img src="images/binaryimage.png" align="center" height="400" alt="binary image"/> +Refining the filtration + The first filtration for the image has 40 terms. One may wish to investigate a filtration with more terms, say 500 terms, with a view to analysing, say, + those 1-cycles that are born by term 25 of the filtration and + that die between terms 50 and 60. The following commands produce the relevant barcode showing that there is precisely one such 1-cycle. + + +<#Include SYSTEM "tutex/4.6d.txt"> + + +

    \beta_0:

    +<img src="images/refinedbc0.gif" align="center" height="100" alt="bar code"/> +

    + \beta_1:

    + <img src="images/refinedbc.gif" align="center" height="200" alt="bar code"/> + + + + Background to the data

    The following image was used in the example. @@ -129,10 +202,10 @@ degree 0 barcode suggest that the image has 20 objects. The degree

    we can load the image as a filtered pure cubical complex F -of filtration length 30 say, and observe the degree zero persistent - Betti numbers to establish that the 21-st term or so of F +of filtration length 40 say, and observe the degree zero persistent + Betti numbers to establish that the 28-th term or so of F seems to be 'noise free' in degree zero. We can then set M equal to - the 21-st term of F and thicken M a couple of + the 28-th term of F and thicken M a couple of times say to remove any tiny holes it may have. We can then construct the complement C of M. Then we can construct a 'neighbourhood thickening' filtration @@ -148,10 +221,10 @@ of filtration length 30 say, and observe the degree zero persistent

    - The pure cubical complex M has the correct number of path - components, namely 25, but its path components are very much subsets of the regions in the image corresponding to coins. The complex M can be thickened repeatedly, subject to no two path components being allowed to merge, in order to obtain a + The pure cubical complex W:=PureComplexComplement(FiltrationTerm(T,25)) has the correct number of path + components, namely 25, but its path components are very much subsets of the regions in the image corresponding to coins. The complex W can be thickened repeatedly, subject to no two path components being allowed to merge, in order to obtain a more realistic image segmentation with path components corresponding more closely to coins. This is done in the follow commands which use a makeshift - function Basins(L) available tutex/basins.ghere . The commands essentially implement the standard watershed segmentation algorithm but do so by using the language of filtered pure cubical complexes. + function Basins(L) available tutex/basins.ghere . The commands essentially implement a standard watershed segmentation algorithm but do so by using the language of filtered pure cubical complexes. <#Include SYSTEM "tutex/4.6c.txt"> @@ -162,6 +235,100 @@ of filtration length 30 say, and observe the degree zero persistent

    + +
    A third example of digital image segmentation + + The following image is number 3096 in the + https://www2.eecs.berkeley.edu/Research/Projects/CS/vision/bsds/ + BSDS500 database of images + + . + +

    + <img src="images/3096.jpg" align="center" height="200" alt="image 3096 from BSDS500"/> + + +

    A common first step in segmenting such an image is to appropriately threshold the corresponding gradient image. + +

    + <img src="images/3096b.jpg" align="center" height="200" alt="gradient image"/> + + <img src="images/3096points.png" align="center" height="200" alt="thresholded gradient image"/> + +

    +The following commands use the thresholded gradient image to produce an outline of the aeroplane. The outline is a pure cubical complex with one path component and with first Betti number equal to 1. + + +<#Include SYSTEM "tutex/4.6g.txt"> + + + <img src="images/3096final.png" align="center" height="200" alt="outline of aeroplane"/> + + + +

    + +
    Naive example of digital image contour extraction + + The following greyscale image is available from the http://www.ipol.im/pub/art/2014/74/FrechetAndConnectedCompDemo.tgzonline appendix to the paper . + +

    + <img src="images/circularGradient.png" align="center" height="250" alt="circular gradient image"/> + + +

    The following commands produce a picture of contours from this image based on greyscale values. They also produce a picture of just the closed contours (the non-closed contours having been homotopy collapsed to points). + + +<#Include SYSTEM "tutex/4.17.txt"> + + + Contours from the above greyscale image: +

    + <img src="images/contours.png" align="center" height="250" alt="contours image"/> + +

    + + Closed contours from the above greyscale image: +

    + + <img src="images/closedcontours.png" align="center" height="250" alt="closedcontours image"/> + + +

    Very similar results are obtained when applied to the file circularGradientNoise.png, containing noise, available + from + the http://www.ipol.im/pub/art/2014/74/FrechetAndConnectedCompDemo.tgzonline appendix to the paper . + +

    The number of distinct "light sources" in the image can be read from the countours. Alternatively, this number can be read directly from the barcode produced by the following commands. + + +<#Include SYSTEM "tutex/4.18.txt"> + + +

    + <img src="images/bccircularGradient.png" align="center" height="250" alt="closedcontours image"/> + +

    The seventeen bars in the barcode correspond to seventeen light sources. The length of a bar is a measure of the "persistence" of the corresponding light source. A long bar may initially represent a cluster of several + lights whose members may eventually be distinguished from each other + as new bars (or persistent homology classes) are created. + +

    Here the command PersistentBettiNumbersAlt has been used. This command is explained in the following section. + +

    The follwowing commands use a watershed method to partition the digital image into regions, one region per light source. + A makeshift + function Basins(L), available tutex/basins.ghere , is called. + (The efficiency of + the example could be easily improved. For simplicity + it uses generic commands which, in principle, can be applied to cubical or permutarhedral complexes of higher dimensions.) + + + +<#Include SYSTEM "tutex/4.19.txt"> + + +

    + <img src="images/circularGradientSeg.png" align="center" height="250" alt="segmented image"/> + +

    Alternative approaches to computing persistent homology From any sequence X_0 \subset X_1 \subset X_2 \subset \cdots \subset X_T of cellular spaces @@ -192,9 +359,15 @@ can be computed and the persistent homology can be derived from these homology h

    As an illustration we consider a synthetic data set S consisting of 3527 - points sampled, with errors, from an `unknown' manifold M in \mathbb R^3. From such a data set one can associate + points sampled, with errors, from an `unknown' manifold M in \mathbb R^3. + From such a data set one can associate a 3-dimensional - cubical complex X_0 consisting of one unit cube centred on each (suitably scaled) data point. + cubical complex X_0 consisting of one unit cube centred on each (suitably scaled) data point. A visualization of X_0 is shown below. + + +

    +<img src="images/data.png" align="center" height="300" alt="data cloud"/> +

    Given a pure cubical complex X_s we construct X_{s+1} =X_s \cup \{\overline e^3_\lambda\}_{\lambda\in \Lambda} by adding to X_s each closed unit cube \overline e^3_\lambda in @@ -227,6 +400,26 @@ can be computed and the persistent homology can be derived from these homology h <#Include SYSTEM "tutex/4.7b.txt"> + + +Explicit homology generators + + It could be desirable to obtain explicit representatives of the persistent homology generators that "persist" through a + significant sequence of filtration terms. There are two such generators in degree 1 and one such generator in degree 2. The explicit representatives in degree n could consist of an inclusion of + pure cubical complexes Y_n \subset X_{10} for which the incuced homology + homomorphism H_n(Y_n,\mathbb Z) \rightarrow H_n(X_{10},\mathbb Z) is an isomorphism, and for which Y_n is minimal in the sense that its homotopy type changes if any one or more of its top dimensional cells are removed. + Ideally the space Y_n should be "close to the original dataset" X_0. + The following commands first construct an explicit degree 2 homology generator representative + Y_2\subset X_{10} where Y_2 is homotopy equivalent to X_{10}. + They then construct an explicit degree 1 homology generators representative + Y_1\subset X_{10} where Y_1 is homotopy equivalent to a wedge of two circles. + The final command displays the homology generators representative Y_1. + +<#Include SYSTEM "tutex/4.7c.txt"> + +

    +<img src="images/cubicaltorusgens.png" align="center" height="200" alt="first homology generators"/> +

    @@ -244,7 +437,11 @@ can be computed and the persistent homology can be derived from these homology h

    -<img src="images/1v2x.gif" align="center" height="500" alt="a protein backbone"/> + <img src="images/1v2x.gif" align="center" height="500" alt="a protein backbone"/> + + +

    The next command reads in the pdb file for the T.thermophilus 1V2X protein and represents it as a 3-dimensional pure cubical complex K. A resolution of r=5 is chosen and this results in a representation as a subcomplex K of an ambient rectangular box of volume equal to 184\times 186\times 294 unit cubes. The complex K should have the homotopy type of a circle and the protein backbone is a 1-dimenional curve that should lie in K. The final command displays K. @@ -252,9 +449,14 @@ can be computed and the persistent homology can be derived from these homology h <#Include SYSTEM "tutex/4.11.txt"> -

    +

    <img src="images/1v2xcubical.gif" align="center" height="500" alt="pure cubical complex representing a protein backbone"/> + + +

    Next we create a filtered pure cubical complex by repeatedly thickening K. We perform 15 thickenings, each thickening being a term in the filtration. The \beta_1 barcode for the filtration is displayed. This barcode is a descriptor for the geometry of the protein. For current purposes it suffices to note that the first few terms of the filtration have first homology equal to that of a circle. This indicates that the Euclidean coordinates in the pdb file robustly determine some knot. diff --git a/www/SideLinks/About/aboutPersistent.html b/www/SideLinks/About/aboutPersistent.html index 3949993b..582713f4 100644 --- a/www/SideLinks/About/aboutPersistent.html +++ b/www/SideLinks/About/aboutPersistent.html @@ -330,7 +330,7 @@ consider the digital photo.


    + style="width: 499px; height: 376px;" alt="" src="../../../tst/examples/image1.3.2.png">

    The 20 longish lines in the following barcode for the degree 0 persistent homology correspond to the 20 objects in the photo. The 14 @@ -347,7 +347,7 @@ gap> -F:=ReadImageAsFilteredCubicalComplex("nb.png",20);
    +F:=ReadImageAsFilteredCubicalComplex("image1.3.2.png",20);
    Filtered pure cubical complex of dimension 2.

    gap> P0:=PersistentHomologyOfFilteredCubicalComplex(F,0);;
    diff --git a/www/download/downloadContent.html b/www/download/downloadContent.html index c88bf9a3..ed860234 100644 --- a/www/download/downloadContent.html +++ b/www/download/downloadContent.html @@ -19,23 +19,23 @@

    Download Instructions

    • First download the file hap1.61.tar.gz + href="https://github.com/gap-packages/hap/releases/download/v1.62/hap-1.62.tar.gz">hap1.62.tar.gz which contains the most recent development version of HAP to the subdirectory "pkg/" of GAP. If you don't have access to this subdirectory, then create a directory "pkg" in your home directory and download the file there. (If you'd prefer to download the most recent development version of HAP then download the file  hap1.61-dev.tar.gz + href="https://github.com/gap-packages/hap/releases/download/v1.47/hap-1.47.tar.gz">hap1.62-dev.tar.gz instead.)
    • Change to directory "pkg/" and type "gunzip -hap1.61.tar.gz" +hap1.62.tar.gz" followed by "tar --xvf hap1.61.tar" .
      • +-xvf hap1.62.tar" .
    • Start GAP. (If you have created "pkg" in your home @@ -51,7 +51,7 @@

      Download Instructions

    • Help on HAP can be found on the HAP home page (a version of which is included in directory -"pkg/Hap1.61/www" of the distribution).
      • +"pkg/Hap1.62/www" of the distribution).
    • A few of HAP's (optional) functions rely on Polymake @@ -67,7 +67,7 @@

      Download Instructions

    • Performance can be improved by using a compiled version of the HAP library. The following steps will produce a compiled version.
      -(1) Change to the directory "pkg/Hap1.61/" .
      +(1) Change to the directory "pkg/Hap1.62/" .
      (2) Edit the file "compile" so that: PKGDIR is equal to the path to the
      directory "pkg" where your GAP packages are stored; GACDIR is equal to the
      @@ -78,7 +78,7 @@

      Download Instructions

      • Should you want to return to an uncompiled version, change to the directory
        -"pkg/Hap1.61/" and type "./uncompile".
        • +"pkg/Hap1.62/" and type "./uncompile".
    diff --git a/www/home/content.html b/www/home/content.html index 7df15d52..79a8bdb7 100644 --- a/www/home/content.html +++ b/www/home/content.html @@ -15,8 +15,8 @@ algebra library for use with the GAP computer algebra system, and is still under development. The current version hap1.61.tar.gz was released on -02 Jan 2024.

    + href="https://github.com/gap-packages/hap/releases/download/v1.62/hap-1.62.tar.gz">hap1.62.tar.gz was released on +01 Feb 2024.