<Chapter><Heading>Bredon homology</Heading>
<Section><Heading>Davis complex</Heading>

<P/>The following example computes the Bredon homology
<P/><M>\underline H_0(W,{\cal R}) = \mathbb Z^{21}</M>
<P/> for the infinite Coxeter group <M>W</M> associated to the Dynkin diagram shown in the computation, with coefficients in the complex representation ring.

<Example>
<#Include SYSTEM "tutex/8.1.txt">
</Example>
<Alt Only="HTML">&lt;img src="images/infcoxdiag.gif" align="center" height="160" alt="Coxeter diagram"/>

</Alt>

<Example>
<#Include SYSTEM "tutex/8.2.txt">
</Example>

</Section>
<Section><Heading>Arithmetic groups</Heading>

<P/>The following example computes the Bredon homology
<P/><M>\underline H_0(SL_2({\cal O}_{-3}),{\cal R}) = \mathbb Z_2\oplus \mathbb Z^{9}</M>
<P/><M>\underline H_1(SL_2({\cal O}_{-3}),{\cal R}) = \mathbb Z</M>
<P/>for <M>{\cal O}_{-3}</M> the ring of integers of the number field
<M>\mathbb Q(\sqrt{-3})</M>, and <M>\cal R</M> the complex reflection ring.

<Example>
<#Include SYSTEM "tutex/8.3.txt">
</Example>

</Section>
<Section><Heading>Crystallographic groups</Heading>

<P/>The following example computes the Bredon homology
<P/><M>\underline H_0(G,{\cal R}) = \mathbb Z^{17}</M>
<P/> for <M>G</M> the second crystallographic group of dimension <M>4</M> in
<B>GAP</B>'s library of crystallographic groups, and for <M>\cal R</M> the Burnside ring.

<Example>
<#Include SYSTEM "tutex/8.4.txt">
</Example>

</Section>
</Chapter>
