The following example concerns the crossed module
∂: G→ Aut(G), g↦ (x↦ gxg^-1)
associated to the dihedral group G of order 16. This crossed module represents, up to homotopy type, a connected space X with π_iX=0 for ige 3, π_2X=Z(G), π_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, Z)= Z_2 ⊕ Z_2,
H_2(X, Z)= Z_2,
H_3(X, Z)= Z_2 ⊕ Z_2 ⊕ Z_2,
H_4(X, Z)= Z_2 ⊕ Z_2 ⊕ Z_2,
H_5(X, Z)= Z_2 ⊕ Z_2 ⊕ Z_2 ⊕ Z_2⊕ Z_2⊕ 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 ]
The following example concerns the Eilenberg-MacLane space X=K( Z,3) which is a path-connected space with π_3X= Z, π_iX=0 for 3ne ige 1. This space is represented by a simplicial group, and perturbation techniques are used to compute
H_7(X, Z)= 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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