From e5306a11e25202d4f088d362dfc7f879ec62e1e5 Mon Sep 17 00:00:00 2001 From: grahamknockillaree Date: Fri, 12 Jan 2024 20:43:32 +0000 Subject: [PATCH] Add files via upload --- tutorial/chap11.html | 25 ++++++++++++++++++++++++- tutorial/chap11.txt | 32 ++++++++++++++++++++++++++++++-- tutorial/chap11_mj.html | 25 ++++++++++++++++++++++++- 3 files changed, 78 insertions(+), 4 deletions(-) diff --git a/tutorial/chap11.html b/tutorial/chap11.html index 42d66b6a..24579770 100644 --- a/tutorial/chap11.html +++ b/tutorial/chap11.html @@ -138,7 +138,7 @@

11.2 Resolutions for very small finite groups -

The suspicion that this resolution R_∗ is periodic of period 4 can be verified by constructing the chain complex C_∗=R_∗⊗_ Z ZG and verifying that boundary matrices repeat with period 4.

+

The suspicion that this resolution R_∗ is periodic of period 4 can be confirmed by constructing the chain complex C_∗=R_∗⊗_ Z ZG and verifying that boundary matrices repeat with period 4.

A second example of a periodic resolution, for the Dihedral group D_2k+1=⟨ x, y | x^2= xy^kx^-1y^-k-1⟩ of order 2k+2 in the case k=1, is constructed and verified for periodicity in the next example.

@@ -176,6 +176,29 @@

11.2 Resolutions for very small finite groups +

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.

+ + +
+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 ]
+
+
+

11.3 Resolutions for finite groups acting on orbit polytopes

diff --git a/tutorial/chap11.txt b/tutorial/chap11.txt index f9d06d44..8182574a 100644 --- a/tutorial/chap11.txt +++ b/tutorial/chap11.txt @@ -41,8 +41,8 @@  The suspicion that this resolution R_∗ is periodic of period 4 can be - verified by constructing the chain complex C_∗=R_∗⊗_ Z ZG and verifying that - boundary matrices repeat with period 4. + confirmed by constructing the chain complex C_∗=R_∗⊗_ Z ZG and verifying + that boundary matrices repeat with period 4. A second example of a periodic resolution, for the Dihedral group D_2k+1=⟨ x, y | x^2= xy^kx^-1y^-k-1⟩ of order 2k+2 in the case k=1, is constructed @@ -88,6 +88,34 @@   + 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. + +  Example  + 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); +  + 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 ] +  +  + 11.3 Resolutions for finite groups acting on orbit polytopes diff --git a/tutorial/chap11_mj.html b/tutorial/chap11_mj.html index acd7d771..28072648 100644 --- a/tutorial/chap11_mj.html +++ b/tutorial/chap11_mj.html @@ -141,7 +141,7 @@

11.2 Resolutions for very small finite groups -

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.

@@ -179,6 +179,29 @@

11.2 Resolutions for very small finite groups +

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 \(4\).

+ + +
+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 ]
+
+
+

11.3 Resolutions for finite groups acting on orbit polytopes