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The cohomology of a group G with coefficients in a ZG-module A is defined as:
Hn(G,A) = | Ker( HomZG(Rn,A) → HomZG(Rn+1,A) ) Image( HomZG(Rn-1,A) → HomZG(Rn,A)
When the abelian group underlying A is free of rank n we can encode A as a group homomorphism A:G → GLn(Z). When G is a permutation group of degree n the free abelian group Zn admits a canonical G-action defined by g·(x1, x2, ... , xn) = (xg'(1) , xg'(2) , ... , xg'(n)) where g'=g-1, for g in G and xi in Z. This canonical permutation module A can be constructed for any permutation group G using the HAP command PermToMatrixGroup(). For example:
gap> G:=AlternatingGroup(5);; gap> A:=PermToMatrixGroup(G,5); [ (1,2,3,4,5), (3,4,5) ] -> [ [ [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ], [ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ], [ 1, 0, 0, 0, 0 ] ], [ [ 1, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ], [ 0, 0, 1, 0, 0 ] ] ]
The following commands show that: the 6th cohomology of the alternating group G=A5 with coefficients in its 5-dimensional canonical permutation module A is H6(G,A) = Z2+Z6. The 3rd cohomology of the even subgroup B+ of the 5-string Braid group, again with coefficients in the permutation module A (considered as a B+-module via the quotient homomorphism B+ → A5) is H3(B+,A) = Z2+Z6+Z3.
gap> Alt5:=AlternatingGroup(5);; gap> A:=PermToMatrixGroup(SymmetricGroup(5),5);; gap> R:=ResolutionFiniteGroup(Alt5,7);; gap> TR:=HomToIntegralModule(R,A);; gap> Cohomology(TR,6); [ 2, 6 ] gap> D:=[[1,[2,3]],[2,[3,3]],[3,[4,3]]];; gap> R:=ResolutionArtinGroup(D,10);; gap> Brd5:=R!.group;; Brd5Gens:=GeneratorsOfGroup(Brd5);; gap> ImGens:=[Image(A,(1,2)),Image(A,(2,3)),Image(A,(3,4)),Image(A,(4,5))];; gap> B:=GroupHomomorphismByImages(Brd5,Image(A),Brd5Gens,ImGens);; gap> EvBrd5:=EvenSubgroup(Brd5);; gap> S:=ResolutionSubgroup(R,EvBrd5);; gap> TS:=HomToIntegralModule(S,B);; gap> Cohomology(TS,3); [ 2, 6, 0, 0, 0 ]
Such commands can be used, for instance, to produce strong evidence in support of the following.
### Conjecture: Let Z(n) denote the free abelian group of rank n endowed with the canonical permutation action of the symmetric group Sn. Let Z be the free abelian group of rank 1 with trivial action of Sn. There is an isomorphism Hk(Sn, Z(n)) = Hk(Sn-1, Z) for all n>1 and for all k.
The following commands support the conjecture. (It might be possible to use the techniques of [M. Salvetti, "Cohomology of Coxeter groups", Topology and its Applications 118 (2002), 199-208] to prove the conjecture.)
gap> R:=[];;C:=[];;D:=[];; gap> for n in [2..5] do > R[n]:=ResolutionFiniteGroup(SymmetricGroup(n),6); > C[n]:=HomToIntegralModule(R[n],PermToMatrixGroup(SymmetricGroup(n),n)); > D[n]:=HomToIntegers(R[n]); > od; gap> for k in [0..5] do > for n in [2..4] do > Print("Twisted Cohomology: H",k,"(S",n+1,") = ",Cohomology(C[n+1],k),"\n"); > Print("Integral Cohomology: H",k,"(S",n,") = ",Cohomology(D[n],k),"\n\n"); > od;od; Twisted Cohomology: H0(S3) = [ 0 ] Integral Cohomology: H0(S2) = [ 0 ] Twisted Cohomology: H0(S4) = [ 0 ] Integral Cohomology: H0(S3) = [ 0 ] Twisted Cohomology: H0(S5) = [ 0 ] Integral Cohomology: H0(S4) = [ 0 ] Twisted Cohomology: H1(S3) = [ ] Integral Cohomology: H1(S2) = [ ] Twisted Cohomology: H1(S4) = [ ] Integral Cohomology: H1(S3) = [ ] Twisted Cohomology: H1(S5) = [ ] Integral Cohomology: H1(S4) = [ ] Twisted Cohomology: H2(S3) = [ 2 ] Integral Cohomology: H2(S2) = [ 2 ] Twisted Cohomology: H2(S4) = [ 2 ] Integral Cohomology: H2(S3) = [ 2 ] Twisted Cohomology: H2(S5) = [ 2 ] Integral Cohomology: H2(S4) = [ 2 ] Twisted Cohomology: H3(S3) = [ ] Integral Cohomology: H3(S2) = [ ] Twisted Cohomology: H3(S4) = [ ] Integral Cohomology: H3(S3) = [ ] Twisted Cohomology: H3(S5) = [ 2 ] Integral Cohomology: H3(S4) = [ 2 ] Twisted Cohomology: H4(S3) = [ 2 ] Integral Cohomology: H4(S2) = [ 2 ] Twisted Cohomology: H4(S4) = [ 6 ] Integral Cohomology: H4(S3) = [ 6 ] Twisted Cohomology: H4(S5) = [ 2, 12 ] Integral Cohomology: H4(S4) = [ 2, 12 ] Twisted Cohomology: H5(S3) = [ ] Integral Cohomology: H5(S2) = [ ] Twisted Cohomology: H5(S4) = [ ] Integral Cohomology: H5(S3) = [ ] Twisted Cohomology: H5(S5) = [ 2 ] Integral Cohomology: H5(S4) = [ 2 ]
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