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9 Simplicial groups
 9.1 Crossed modules
 9.2 Eilenberg-MacLane spaces

9 Simplicial groups

9.1 Crossed modules

The following example concerns the crossed module

\(\partial\colon G\rightarrow Aut(G), g\mapsto (x\mapsto gxg^{-1})\)

associated to the dihedral group \(G\) of order \(16\). This crossed module represents, up to homotopy type, a connected space \(X\) with \(\pi_iX=0\) for \(i\ge 3\), \(\pi_2X=Z(G)\), \(\pi_1X = Aut(G)/Inn(G)\). The space \(X\) can be represented, up to homotopy, by a simplicial group. That simplicial group is used in the example to compute

\(H_1(X,\mathbb Z)= \mathbb Z_2 \oplus \mathbb Z_2\),

\(H_2(X,\mathbb Z)= \mathbb Z_2 \),

\(H_3(X,\mathbb Z)= \mathbb Z_2 \oplus \mathbb Z_2 \oplus \mathbb Z_2\),

\(H_4(X,\mathbb Z)= \mathbb Z_2 \oplus \mathbb Z_2 \oplus \mathbb Z_2\),

\(H_5(X,\mathbb Z)= \mathbb Z_2 \oplus \mathbb Z_2 \oplus \mathbb Z_2 \oplus \mathbb Z_2\oplus \mathbb Z_2\oplus \mathbb Z_2\).

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

gap> C:=AutomorphismGroupAsCatOneGroup(DihedralGroup(16));
Cat-1-group with underlying group Group( 
[ f1, f2, f3, f4, f5, f6, f7, f8, f9 ] ) . 

gap> Size(C);
512
gap> Q:=QuasiIsomorph(C);
Cat-1-group with underlying group Group( [ f9, f8, f1, f2*f3, f5 ] ) . 

gap> Size(Q);
32

gap> N:=NerveOfCatOneGroup(Q,6);
Simplicial group of length 6

gap> K:=ChainComplexOfSimplicialGroup(N);
Chain complex of length 6 in characteristic 0 . 

gap> Homology(K,1);
[ 2, 2 ]
gap> Homology(K,2);
[ 2 ]
gap> Homology(K,3);
[ 2, 2, 2 ]
gap> Homology(K,4);
[ 2, 2, 2 ]
gap> Homology(K,5);
[ 2, 2, 2, 2, 2, 2 ]

9.2 Eilenberg-MacLane spaces

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

\(H_7(X,\mathbb Z)=\mathbb Z_3\).

gap> A:=AbelianPcpGroup([0]);;AbelianInvariants(A);
[ 0 ]
gap> K:=EilenbergMacLaneSimplicialGroup(A,3,8);
Simplicial group of length 8

gap> C:=ChainComplexOfSimplicialGroup(K);
Chain complex of length 8 in characteristic 0 . 

gap> Homology(C,7);
[ 3 ]

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