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25 changes: 24 additions & 1 deletion tutorial/chap11.html
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Expand Up @@ -138,7 +138,7 @@ <h4>11.2 <span class="Heading">Resolutions for very small finite groups</span></

</pre></div>

<p>The suspicion that this resolution <span class="SimpleMath">R_∗</span> is periodic of period <span class="SimpleMath">4</span> can be verified by constructing the chain complex <span class="SimpleMath">C_∗=R_∗⊗_ Z ZG</span> and verifying that boundary matrices repeat with period <span class="SimpleMath">4</span>.</p>
<p>The suspicion that this resolution <span class="SimpleMath">R_∗</span> is periodic of period <span class="SimpleMath">4</span> can be confirmed by constructing the chain complex <span class="SimpleMath">C_∗=R_∗⊗_ Z ZG</span> and verifying that boundary matrices repeat with period <span class="SimpleMath">4</span>.</p>

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

Expand Down Expand Up @@ -176,6 +176,29 @@ <h4>11.2 <span class="Heading">Resolutions for very small finite groups</span></

</pre></div>

<p>The performance of the function <code class="code">ResolutionSmallGroup(G,n)</code> is very sensistive to the choice of presentation for the input group <span class="SimpleMath">G</span>. If <span class="SimpleMath">G</span> is an fp-group then the defining presentation for <span class="SimpleMath">G</span> is used. If <span class="SimpleMath">G</span> is a permutaion group or finite matrix group then <strong class="button">GAP</strong> functions are invoked to find a presentation for <span class="SimpleMath">G</span>. The following commands use a geometrically derived presentation for <span class="SimpleMath">SL(2,5)</span> as input in order to obtain the first few terms of a periodic resolution for this group of period <span class="SimpleMath">4</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Y:=PoincareDodecahedronCWComplex( </span>
<span class="GAPprompt">&gt;</span> <span class="GAPinput">[[1,2,3,4,5],[6,7,8,9,10]],</span>
<span class="GAPprompt">&gt;</span> <span class="GAPinput">[[1,11,16,12,2],[19,9,8,18,14]],</span>
<span class="GAPprompt">&gt;</span> <span class="GAPinput">[[2,12,17,13,3],[20,10,9,19,15]],</span>
<span class="GAPprompt">&gt;</span> <span class="GAPinput">[[3,13,18,14,4],[16,6,10,20,11]],</span>
<span class="GAPprompt">&gt;</span> <span class="GAPinput">[[4,14,19,15,5],[17,7,6,16,12]],</span>
<span class="GAPprompt">&gt;</span> <span class="GAPinput">[[5,15,20,11,1],[18,8,7,17,13]]);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">G:=FundamentalGroup(Y);</span>
&lt;fp group on the generators [ f1, f2 ]&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">RelatorsOfFpGroup(G);</span>
[ f2^-1*f1^-1*f2*f1^-1*f2^-1*f1, f2^-1*f1*f2^2*f1*f2^-1*f1^-1 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">StructureDescription(G);</span>
"SL(2,5)"
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">R:=ResolutionSmallGroup(G,3);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">List([0..3],R!.dimension); </span>
[ 1, 2, 2, 1 ]

</pre></div>

<p><a id="X86C0983E81F706F5" name="X86C0983E81F706F5"></a></p>

<h4>11.3 <span class="Heading">Resolutions for finite groups acting on orbit polytopes</span></h4>
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32 changes: 30 additions & 2 deletions tutorial/chap11.txt
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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
Expand Down Expand Up @@ -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);
<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

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25 changes: 24 additions & 1 deletion tutorial/chap11_mj.html
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Expand Up @@ -141,7 +141,7 @@ <h4>11.2 <span class="Heading">Resolutions for very small finite groups</span></

</pre></div>

<p>The suspicion that this resolution <span class="SimpleMath">\(R_\ast\)</span> is periodic of period <span class="SimpleMath">\(4\)</span> can be verified by constructing the chain complex <span class="SimpleMath">\(C_\ast=R_\ast\otimes_{\mathbb Z}\mathbb ZG\)</span> and verifying that boundary matrices repeat with period <span class="SimpleMath">\(4\)</span>.</p>
<p>The suspicion that this resolution <span class="SimpleMath">\(R_\ast\)</span> is periodic of period <span class="SimpleMath">\(4\)</span> can be confirmed by constructing the chain complex <span class="SimpleMath">\(C_\ast=R_\ast\otimes_{\mathbb Z}\mathbb ZG\)</span> and verifying that boundary matrices repeat with period <span class="SimpleMath">\(4\)</span>.</p>

<p>A second example of a periodic resolution, for the Dihedral group <span class="SimpleMath">\(D_{2k+1}=\langle x, y\ |\ x^2= xy^kx^{-1}y^{-k-1}\rangle\)</span> of order <span class="SimpleMath">\(2k+2\)</span> in the case <span class="SimpleMath">\(k=1\)</span>, is constructed and verified for periodicity in the next example.</p>

Expand Down Expand Up @@ -179,6 +179,29 @@ <h4>11.2 <span class="Heading">Resolutions for very small finite groups</span></

</pre></div>

<p>The performance of the function <code class="code">ResolutionSmallGroup(G,n)</code> is very sensistive to the choice of presentation for the input group <span class="SimpleMath">\(G\)</span>. If <span class="SimpleMath">\(G\)</span> is an fp-group then the defining presentation for <span class="SimpleMath">\(G\)</span> is used. If <span class="SimpleMath">\(G\)</span> is a permutaion group or finite matrix group then <strong class="button">GAP</strong> functions are invoked to find a presentation for <span class="SimpleMath">\(G\)</span>. The following commands use a geometrically derived presentation for <span class="SimpleMath">\(SL(2,5)\)</span> as input in order to obtain the first few terms of a periodic resolution for this group of period <span class="SimpleMath">\(4\)</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Y:=PoincareDodecahedronCWComplex( </span>
<span class="GAPprompt">&gt;</span> <span class="GAPinput">[[1,2,3,4,5],[6,7,8,9,10]],</span>
<span class="GAPprompt">&gt;</span> <span class="GAPinput">[[1,11,16,12,2],[19,9,8,18,14]],</span>
<span class="GAPprompt">&gt;</span> <span class="GAPinput">[[2,12,17,13,3],[20,10,9,19,15]],</span>
<span class="GAPprompt">&gt;</span> <span class="GAPinput">[[3,13,18,14,4],[16,6,10,20,11]],</span>
<span class="GAPprompt">&gt;</span> <span class="GAPinput">[[4,14,19,15,5],[17,7,6,16,12]],</span>
<span class="GAPprompt">&gt;</span> <span class="GAPinput">[[5,15,20,11,1],[18,8,7,17,13]]);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">G:=FundamentalGroup(Y);</span>
&lt;fp group on the generators [ f1, f2 ]&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">RelatorsOfFpGroup(G);</span>
[ f2^-1*f1^-1*f2*f1^-1*f2^-1*f1, f2^-1*f1*f2^2*f1*f2^-1*f1^-1 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">StructureDescription(G);</span>
"SL(2,5)"
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">R:=ResolutionSmallGroup(G,3);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">List([0..3],R!.dimension); </span>
[ 1, 2, 2, 1 ]

</pre></div>

<p><a id="X86C0983E81F706F5" name="X86C0983E81F706F5"></a></p>

<h4>11.3 <span class="Heading">Resolutions for finite groups acting on orbit polytopes</span></h4>
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