<Chapter><Heading>Simplicial groups</Heading>
<Section><Heading>Crossed modules</Heading>

The following example concerns the crossed module
<P/><M>\partial\colon G\rightarrow Aut(G), g\mapsto (x\mapsto gxg^{-1})</M>
<P/>associated to the dihedral group <M>G</M> of order <M>16</M>. This crossed module represents, up to homotopy type,
 a connected space <M>X</M> with <M>\pi_iX=0</M> for <M>i\ge 3</M>,
<M>\pi_2X=Z(G)</M>, <M>\pi_1X = Aut(G)/Inn(G)</M>.
The space <M>X</M> can be represented, up to homotopy, by a simplicial group.
That simplicial group is used in the example to compute
<P/><M>H_1(X,\mathbb Z)= \mathbb Z_2 \oplus \mathbb Z_2</M>,
<P/><M>H_2(X,\mathbb Z)= \mathbb Z_2 </M>,
<P/><M>H_3(X,\mathbb Z)= \mathbb Z_2 \oplus \mathbb Z_2 \oplus \mathbb Z_2</M>,
<P/><M>H_4(X,\mathbb Z)= \mathbb Z_2 \oplus \mathbb Z_2 \oplus \mathbb Z_2</M>,
<P/><M>H_5(X,\mathbb Z)= \mathbb Z_2 \oplus \mathbb Z_2 \oplus \mathbb Z_2 \oplus \mathbb Z_2\oplus \mathbb Z_2\oplus \mathbb Z_2</M>.

<P/>The simplicial group is obtained by viewing the crossed module as a crossed complex and using a nonabelian version of the Dold-Kan theorem.



<Example>
<#Include SYSTEM "tutex/9.1.txt">
</Example>
</Section>

<Section><Heading>Eilenberg-MacLane spaces</Heading>
<P/>The following example concerns the Eilenberg-MacLane space <M>X=K(\mathbb Z,3)</M> which is a path-connected space with  <M>\pi_3X=\mathbb Z</M>,
  <M>\pi_iX=0</M> for <M>3\ne  i\ge 1</M>. This space is represented by a simplicial group, and perturbation techniques are used to compute
<P/><M>H_7(X,\mathbb Z)=\mathbb Z_3</M>.

<Example>
<#Include SYSTEM "tutex/9.2.txt">
</Example>


</Section>
</Chapter>
