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<div class="ChapSects"><a href="chap9_mj.html#X786DB80A8693779E">9 <span class="Heading">Bredon homology</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X7B0212F97F3D442A">9.1 <span class="Heading">Davis complex</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X7AFFB32587D047FE">9.2 <span class="Heading">Arithmetic groups</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X7DEBF2BB7D1FB144">9.3 <span class="Heading">Crystallographic groups</span></a>
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<h3>9 <span class="Heading">Bredon homology</span></h3>

<p><a id="X7B0212F97F3D442A" name="X7B0212F97F3D442A"></a></p>

<h4>9.1 <span class="Heading">Davis complex</span></h4>

<p>The following example computes the Bredon homology</p>

<p><span class="SimpleMath">\(\underline H_0(W,{\cal R}) = \mathbb Z^{21}\)</span></p>

<p>for the infinite Coxeter group <span class="SimpleMath">\(W\)</span> associated to the Dynkin diagram shown in the computation, with coefficients in the complex representation ring.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">D:=[[1,[2,3]],[2,[3,3]],[3,[4,3]],[4,[5,6]]];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">CoxeterDiagramDisplay(D);</span>

</pre></div>

<p><img src="images/infcoxdiag.gif" align="center" height="160" alt="Coxeter diagram"/></p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">C:=DavisComplex(D);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">D:=TensorWithComplexRepresentationRing(C);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(D,0);</span>
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]

</pre></div>

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<h4>9.2 <span class="Heading">Arithmetic groups</span></h4>

<p>The following example computes the Bredon homology</p>

<p><span class="SimpleMath">\(\underline H_0(SL_2({\cal O}_{-3}),{\cal R}) = \mathbb Z_2\oplus \mathbb Z^{9}\)</span></p>

<p><span class="SimpleMath">\(\underline H_1(SL_2({\cal O}_{-3}),{\cal R}) = \mathbb Z\)</span></p>

<p>for <span class="SimpleMath">\({\cal O}_{-3}\)</span> the ring of integers of the number field <span class="SimpleMath">\(\mathbb Q(\sqrt{-3})\)</span>, and <span class="SimpleMath">\(\cal R\)</span> the complex reflection ring.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">R:=ContractibleGcomplex("SL(2,O-3)");;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRigid(R);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">S:=BaryCentricSubdivision(R);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRigid(S);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">C:=TensorWithComplexRepresentationRing(S);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(C,0);</span>
[ 2, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(C,1);</span>
[ 0 ]

</pre></div>

<p><a id="X7DEBF2BB7D1FB144" name="X7DEBF2BB7D1FB144"></a></p>

<h4>9.3 <span class="Heading">Crystallographic groups</span></h4>

<p>The following example computes the Bredon homology</p>

<p><span class="SimpleMath">\(\underline H_0(G,{\cal R}) = \mathbb Z^{17}\)</span></p>

<p>for <span class="SimpleMath">\(G\)</span> the second crystallographic group of dimension <span class="SimpleMath">\(4\)</span> in <strong class="button">GAP</strong>'s library of crystallographic groups, and for <span class="SimpleMath">\(\cal R\)</span> the Burnside ring.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=SpaceGroup(4,2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">gens:=GeneratorsOfGroup(G);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:=CrystGFullBasis(G);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R:=CrystGcomplex(gens,B,1);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRigid(R);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">S:=CrystGcomplex(gens,B,0);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRigid(S);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">D:=TensorWithBurnsideRing(S);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(D,0);</span>
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]

</pre></div>


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