GAP (HAP commands) - Chapter 9: Simplicial groups

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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

∂: 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 ]

#### 9.2 Eilenberg-MacLane spaces

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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