GAP (HAP commands) - Chapter 6: Cohomology of groups

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6 Cohomology of groups
6.1 Finite groups
6.2 Nilpotent groups
6.3 Crystallographic groups
6.4 Arithmetic groups
6.5 Artin groups
6.6 Graphs of groups
6.7 Cohomology with coefficients in a module
6.8 Exact cohomology coefficient sequence

### 6 Cohomology of groups

#### 6.1 Finite groups

It is possible to compute the low degree (co)homology of a finite group or monoid of small order directly from the bar resolution. The following commands take this approach to computing the fifth integral homology

\(H_5(Q_4,\mathbb Z) = \mathbb Z_2\oplus\mathbb Z_2\)

of the quaternion group \(G=Q_4\) of order \(8\).

    gap> Q:=QuaternionGroup(8);;
    gap> B:=BarComplexOfMonoid(Q,6);;
    gap> C:=ContractedComplex(B);;
    gap> Homology(C,5);
    [ 2, 2 ]

However, this approach is of limited applicability since the bar resolution involves \(|G|^k\) free generators in degree \(k\). A range of techniques, tailored to specific classes of groups, can be used to compute the (co)homology of larger finite groups.

The following example computes the fourth integral cohomomogy of the Mathieu group \(M_{24}\).

\(H^4(M_{24},\mathbb Z) = \mathbb Z_{12}\)

    gap> GroupCohomology(MathieuGroup(24),4);
    [ 4, 3 ]

The following example computes the third integral homology of the Weyl group \(W=Weyl(E_8)\), a group of order \(696729600\).

\(H_3(Weyl(E_8),\mathbb Z) = \mathbb Z_2 \oplus \mathbb Z_2 \oplus \mathbb Z_{12}\)

    p> L:=SimpleLieAlgebra("E",8,Rationals);;
    gap> W:=WeylGroup(RootSystem(L));;
    gap> Order(W);
    696729600
    gap> GroupHomology(W,3);
    [ 2, 2, 4, 3 ]

The preceding calculation could be achieved more quickly by noting that \(W=Weyl(E_8)\) is a Coxeter group, and by using the associated Coxeter polytope. The following example uses this approach to compute the fourth integral homology of \(W\). It begins by displaying the Coxeter diagram of \(W\), and then computes

\(H_4(Weyl(E_8),\mathbb Z) = \mathbb Z_2 \oplus \mathbb Z_2 \oplus Z_2 \oplus \mathbb Z_2\).

    gap> D:=[[1,[2,3]],[2,[3,3]],[3,[4,3],[5,3]],[5,[6,3]],[6,[7,3]],[7,[8,3]]];;
    gap> CoxeterDiagramDisplay(D);

    gap> polytope:=CoxeterComplex_alt(D,5);;
    gap> R:=FreeGResolution(polytope,5);
    Resolution of length 5 in characteristic 0 for <matrix group with
    8 generators> .
    No contracting homotopy available.

    gap> C:=TensorWithIntegers(R);
    Chain complex of length 5 in characteristic 0 .

    gap> Homology(C,4);
    [ 2, 2, 2, 2 ]

The following example computes the sixth mod-\(2\) homology of the Sylow \(2\)-subgroup \(Syl_2(M_{24})\) of the Mathieu group \(M_{24}\).

\(H_6(Syl_2(M_{24}),\mathbb Z_2) = \mathbb Z_2^{143}\)

    gap> GroupHomology(SylowSubgroup(MathieuGroup(24),2),6,2);
    [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
      2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
      2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
      2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
      2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
      2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
      2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]

The following example constructs the Poincare polynomial

\(p(x)=\frac{1}{-x^3+3*x^2-3*x+1}\)

for the cohomology \(H^\ast(Syl_2(M_{12},\mathbb F_2)\). The coefficient of \(x^n\) in the expansion of \(p(x)\) is equal to the dimension of the vector space \(H^n(Syl_2(M_{12},\mathbb F_2)\). The computation involves Singular's Groebner basis algorithms and the Lyndon-Hochschild-Serre spectral sequence.

    gap> G:=SylowSubgroup(MathieuGroup(12),2);;
    gap> PoincareSeriesLHS(G);
    (1)/(-x_1^3+3*x_1^2-3*x_1+1)

The following example constructs the polynomial

\(p(x)=\frac{x^4-x^3+x^2-x+1}{x^6-x^5+x^4-2*x^3+x^2-x+1}\)

whose coefficient of \(x^n\) is equal to the dimension of the vector space \(H^n(M_{11},\mathbb F_2)\) for all \(n\) in the range \(0\le n\le 14\). The coefficient is not guaranteed correct for \(n\ge 15\).

    gap> PoincareSeriesPrimePart(MathieuGroup(11),2,14);
    (x_1^4-x_1^3+x_1^2-x_1+1)/(x_1^6-x_1^5+x_1^4-2*x_1^3+x_1^2-x_1+1)

#### 6.2 Nilpotent groups

The following example computes

\(H_4(N,\mathbb Z) = \mathbb (Z_3)^4 \oplus \mathbb Z^{84}\)

for the free nilpotent group \(N\) of class \(2\) on four generators.

    gap> F:=FreeGroup(4);; N:=NilpotentQuotient(F,2);;
    gap> GroupHomology(N,4);
    [ 3, 3, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]

#### 6.3 Crystallographic groups

The following example computes

\(H_5(G,\mathbb Z) = \mathbb Z_2 \oplus \mathbb Z_2\)

for the \(3\)-dimensional crystallographic space group \(G\) with Hermann-Mauguin symbol "P62"

    gap> GroupHomology(SpaceGroupBBNWZ("P62"),5);
    [ 2, 2 ]

#### 6.4 Arithmetic groups

The following example computes

\(H_6(SL_2({\cal O},\mathbb Z) = \mathbb Z_2\)

for \({\cal O}\) the ring of integers of the number field \(\mathbb Q(\sqrt{-2})\).

    gap> C:=ContractibleGcomplex("SL(2,O-2)");;
    gap> R:=FreeGResolution(C,7);;
    gap> Homology(TensorWithIntegers(R),6);
    [ 2, 12 ]

#### 6.5 Artin groups

The following example computes

\(H_5(G,\mathbb Z) = \mathbb Z_3\)

for \(G\) the classical braid group on eight strings.

    gap> D:=[[1,[2,3]],[2,[3,3]],[3,[4,3]],[4,[5,3]],[5,[6,3]],[6,[7,3]]];;
    gap> CoxeterDiagramDisplay(D);;

    gap> R:=ResolutionArtinGroup(D,6);;
    gap> C:=TensorWithIntegers(R);;
    gap> Homology(C,5);
    [ 3 ]

#### 6.6 Graphs of groups

The following example computes

\(H_5(G,\mathbb Z) = \mathbb Z_2\oplus Z_2\oplus Z_2 \oplus Z_2 \oplus Z_2\)

for \(G\) the graph of groups corresponding to the amalgamated product \(G=S_5*_{S_3}S_4\) of the symmetric groups \(S_5\) and \(S_4\) over the canonical subgroup \(S_3\).

    gap> S5:=SymmetricGroup(5);SetName(S5,"S5");
    gap> S4:=SymmetricGroup(4);SetName(S4,"S4");
    gap> A:=SymmetricGroup(3);SetName(A,"S3");
    gap> AS5:=GroupHomomorphismByFunction(A,S5,x->x);
    gap> AS4:=GroupHomomorphismByFunction(A,S4,x->x);
    gap> D:=[S5,S4,[AS5,AS4]];
    gap> GraphOfGroupsDisplay(D);

    gap> R:=ResolutionGraphOfGroups(D,6);;
    gap> Homology(TensorWithIntegers(R),5);
    [ 2, 2, 2, 2, 2 ]

#### 6.7 Cohomology with coefficients in a module

There are various ways to represent a \(\mathbb ZG\)-module \(A\) with action \(G\times A \rightarrow A, (g,a)\mapsto \alpha(g,a)\).

One possibility is to use the data type of a \(G\)-Outer Group which involves three components: an \(ActedGroup\) \(A\); an \(Acting Group\) \(G\); a \(Mapping\) \((g,a)\mapsto \alpha(g,a)\). The following example uses this data type to compute the cohomology \(H^4(G,A) =\mathbb Z_5 \oplus \mathbb Z_{10}\) of the symmetric group \(G=S_6\) with coefficients in the integers \(A=\mathbb Z\) where odd permutations act non-trivially on \(A\).

    gap> G:=SymmetricGroup(6);;

    gap> A:=AbelianPcpGroup([0]);;
    gap> alpha:=function(g,a); return a^SignPerm(g); end;;
    gap> A:=GModuleAsGOuterGroup(G,A,alpha);
    ZG-module with abelian invariants [ 0 ] and G= SymmetricGroup( [ 1 .. 6 ] )

    gap> R:=ResolutionFiniteGroup(G,5);;
    gap> C:=HomToGModule(R,A);
    G-cocomplex of length 5 .

    gap> Cohomology(C,4);
    [ 2, 2, 5 ]

If \(A=\mathbb Z^n\) and \(G\) acts as

\(G\times A \rightarrow A, (g, (x_1,x_2,\ldots,x_n)) \mapsto (x_{\pi(g)^{-1}(1)}, x_{\pi(g)^{-1}(2)}, \ldots, x_{\pi(g)^{-1}(n)})\)

where \(\pi\colon G\rightarrow S_n\) is a (not necessarily faithful) permutation representation of degree \(n\) then we can avoid the use of \(G\)-outer groups and use just the homomorphism \(\pi\) instead. The following example uses this data type to compute the cohomology

\(H^6(G,A) =\mathbb Z_2 \oplus \mathbb Z_{6}\)

and the homology

\(H_6(G,A) =\mathbb Z_2 \)

of the alternating group \(G=A_6\) with coefficients in \(A=\mathbb Z^5\) where elements of \(G\) act on \(\mathbb Z^5\) via the canonical permutation of basis elements.

    gap> G:=AlternatingGroup(5);;
    gap> pi:=PermToMatrixGroup(SymmetricGroup(5),5);;
    gap> R:=ResolutionFiniteGroup(G,7);;
    gap> C:=HomToIntegralModule(R,pi);;
    gap> Cohomology(C,6);
    [ 2, 6 ]

    gap> D:=TensorWithIntegralModule(R,pi);;
    gap> Homology(D,6);
    [ 2 ]

#### 6.8 Exact cohomology coefficient sequence

A short exact sequence of \(\mathbb ZG\)-modules \(A \rightarrowtail B \twoheadrightarrow C\) induces a long exact sequence of cohomology groups

\( \rightarrow H^n(G,A) \rightarrow H^n(G,B) \rightarrow H^n(G,C) \rightarrow H^{n+1}(G,A) \rightarrow \) .

Consider the symmetric group \(G=S_4\) and the sequence \( \mathbb Z_4 \rightarrowtail \mathbb Z_8 \twoheadrightarrow \mathbb Z_2\) of trivial \(\mathbb ZG\)-modules. The following commands compute the induced cohomology homomorphism

\(f\colon H^3(S_4,\mathbb Z_4) \rightarrow H^3(S_4,\mathbb Z_8)\)

and determine that the image of this induced homomorphism has order \(8\) and that its kernel has order \(2\).

    gap> G:=SymmetricGroup(4);;
    gap> x:=(1,2,3,4,5,6,7,8);;
    gap> a:=Group(x^2);;
    gap> b:=Group(x);;
    gap> ahomb:=GroupHomomorphismByFunction(a,b,y->y);;
    gap> A:=TrivialGModuleAsGOuterGroup(G,a);;
    gap> B:=TrivialGModuleAsGOuterGroup(G,b);;
    gap> phi:=GOuterGroupHomomorphism();;
    gap> phi!.Source:=A;;
    gap> phi!.Target:=B;;
    gap> phi!.Mapping:=ahomb;;

    gap> Hphi:=CohomologyHomomorphism(phi,3);;

    gap> Size(ImageOfGOuterGroupHomomorphism(Hphi));
    8

    gap> Size(KernelOfGOuterGroupHomomorphism(Hphi));
    2

The following commands then compute the homomorphism

\(H^3(S_4,\mathbb Z_8) \rightarrow H^3(S_4,\mathbb Z_2)\)

induced by \(\mathbb Z_4 \rightarrowtail \mathbb Z_8 \twoheadrightarrow \mathbb Z_2\), and determine that the kernel of this homomorphsim has order \(8\).

    gap> bhomc:=NaturalHomomorphismByNormalSubgroup(b,a);
    gap> B:=TrivialGModuleAsGOuterGroup(G,b);
    gap> C:=TrivialGModuleAsGOuterGroup(G,Image(bhomc));
    gap> psi:=GOuterGroupHomomorphism();
    gap> psi!.Source:=B;
    gap> psi!.Target:=C;
    gap> psi!.Mapping:=bhomc;

    gap> Hpsi:=CohomologyHomomorphism(psi,3);

    gap> Size(KernelOfGOuterGroupHomomorphism(Hpsi));
    8

The following commands then compute the connecting homomorphism

\(H^2(S_4,\mathbb Z_2) \rightarrow H^3(S_4,\mathbb Z_4)\)

and determine that the image of this homomorphism has order \(2\).

    gap> delta:=ConnectingCohomologyHomomorphism(psi,2);;
    gap> Size(ImageOfGOuterGroupHomomorphism(delta));

Note that the various orders are consistent with exactness of the sequence

\(H^2(S_4,\mathbb Z_2) \rightarrow H^3(S_4,\mathbb Z_4) \rightarrow H^3(S_4,\mathbb Z_8) \rightarrow H^3(S_4,\mathbb Z_2) \) .

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