AboutHap

About HAP: Resolutions For Extensions

Consider a direct product of groups E=N×G. Given a free ZN-resolution R of Z, and a free ZG-resolution S of Z, one can construct a free ZE-resolution of Z by forming a tensor product of resolutions R(×)S. This is a special case of a result by Eilenberg and Zilber. In dimension n the tensor product has free generators e(×)f for each i-dimensional generator e in R and each j-dimensional generator f in S with i+j=n. The boundary homomorphism in the tensor product is described by an explicit formula involving just the boundaries of R and S.

The following commands use this tensor product to show that the direct product A₅×S₅of the alternating group of degree 5 with the symmetric group of degree 5 has 6-dimensional integral homology H₆(A₅×S₅,Z)=(Z₂)¹⁰ +Z₆ .

gap> R:=ResolutionFiniteGroup(AlternatingGroup(5),7);;
gap> S:=ResolutionFiniteGroup(SymmetricGroup(5),7);;
gap> RS:=ResolutionDirectProduct(R,S);;
gap> TRS:=TensorWithIntegers(RS);;
gap> Homology(TRS,6);
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6 ]

The tensor product of resolutions can be used to construct a (minimal) resolution for any abelian group. For instance, the following commands confirm the well-known fact that the free abelian group Z⁵ of rank 5 has H₄(Z⁵,Z) = Z⁵.

gap> R:=ResolutionAbelianGroup([0,0,0,0,0],5);;
gap> TR:=TensorWithIntegers(R);;
gap> Homology(TR,4);
[ 0, 0, 0, 0, 0 ]

Consider now the more general situation of a short exact sequence 1 → N → E → G → 1. In [C.T.C. Wall, "Resolutions for extensions of groups", Proc. Cambridge Philos. Soc, 57 (1961), 251-255] it was shown that in this case too there exists a free ZE-resolution with the same number of generators as in the tensor product R(×)S. The boundary homomorphism in the resolution is generally more complicated than that in the tensor product.

A number of related results have been discovered and studied by various people under the general name of Perturbation Theory. One variation of Wall's result is given in [G. Ellis, J. Harris & E. Sköldberg, "Polytopal resolutions for finite groups", J. Reine Angewandte Math.]. It concerns the more general case where N is not necessarily normal in G and also provides explicit formulae for the boundary in the "twisted tensor product". These formulae involve, in addition to the boundaries of R and S, a contracting homotopy on R (by which we mean a family of abelian group homomorphisms h_n:R_n→R_n+1related to the boundary homomorphisms d_n:R_n→R_n-1via the formula d_nh_n-1d_n=d_n). Formulae are also given for a contracting homotopy on the twisted tensor product itself. These formulae can be used to build resolutions and compute homology, as the following commands illustrate.

gap> P:=SylowSubgroup(MathieuGroup(23),2);; gens:=GeneratorsOfGroup(P);;

gap> gensP:=[gens[4], gens[6], gens[2]];; Order(P)=Order(Group(gensP));
true

gap> hom:=NaturalHomomorphismByNormalSubgroup(P,LowerCentralSeries(P)[3]);;

gap> gensG:=List(GensP,x->Image(hom,x));;

gap> S:=ResolutionfFiniteGroup(gensG,6);;

gap> R:=ResolutionFiniteExtension(gensP,gensG,S,6);;

gap> TR:=TensorWithIntegers(R);;

gap> Homology(TR,5);
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]

The function ResolutionFiniteExtension(gensE,gensG,S,n) inputs generators for the group E, the images of these generators in a quotient group G, at least n terms of a ZG-resolution S and a positive integer n. It outputs n terms of a ZE-resolution R. Thus the fifth integral homology of the Sylow 2-subgroup P of M₂₃ is H₅(P,Z)=(Z₂)¹⁴.

The drawback to this divide-and-conquer technique is that it tends to substantially increase the number of generators in the resolution for E. Nevertheless, in low dimensions is can be repeatedly applied to large groups. One function for doing this is ResolutionNormalSeries([L₁, ... , L_k],n). The input is an integer n and a chain of subgroups L₁ > L₂ > ... > L_k each of which is normal in L₁. It outputs n terms of a resolution for the group L₁/L_k.

The following commands show that the biggest Mathieu Group M₂₄ has Sylow 2-subgroup P with sixth mod 2 homology H₆(P,Z₂)=(Z₂)¹⁴³.

gap> P:=SylowSubgroup(MathieuGroup(24),2);;

gap> R:=ResolutionNormalSeries(LowerCentraSeries(P),7);;

gap> TR:=TensorWithIntegersModP(R,2);

gap> Homology(R,6);
143

For an example involving a matrix group consider the group K arising as the kernel of the canonical homomorphism SL₂(Z_p³) --> SL₂(Z_p) with p an odd prime. This group has order p⁶ and exponent p². It was show in [J. Pakianathan, "Exponents and the cohomology of finite groups, Proc. American Math. Soc., Vol. 128, No. 7 (1999), 1893-1897] that the homology groups H_n(K,Z) have exponent less than or equal to p² for all but at most finitely many n. In an earlier and longer paper [W. Browder & J. Pakianathan, "Cohomology of uniformly powerful p-groups", Trans. American Math. Soc., Vol. 352, No. 6 (1999), 2659-2688] it had been proved that H₃(K,Z) has exponent p³. The group K is therefore an example of a p-group for which the maximal exponent of the homology groups H_n(K,Z) occurs only finitely many times, thus disproving a conjecture of A. Adem.

The exponent of the low dimensional homology groups H_n(K,Z), for say p=3, can be computed using the following commands. (A quick way to construct K is as a maximal subgroup of the Sylow 3-subgroup of SL₂(2,Z₂₇).)

gap> P:=SylowSubgroup(SL(2,Integers mod 27),3);;

gap> K:=MaximalSubgroups(P)[2];
<group of 2x2 matrices of size 729 in characteristic 27>

gap> L:=LowerCentralSeries(K);;

gap> R:=ResolutionNormalSeries([L[1],L[2],Group(Identity(K))],6);;

gap> TR:=TensorWithIntegers(R);;

gap> Homology(TR,1);
[ 3, 3, 3 ]

gap> Homology(TR,2);
[3, 3, 3]

gap> Homology(TR,3);
[3, 3, 3, 3, 3, 3, 27]

gap> Homology(TR,4);
[ 3, 3, 3, 3, 3, 3, 3, 3 ]

gap> Homology(TR,5);
[ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 9, 9, 9 ]

The following commands show that the finitely presented dihedral group D₁₀₀ of order 200 has integral homology H₉₉(D₁₀₀,Z)=(Z₂)⁵⁰+Z₁₀₀ in dimension 99.

gap> F:=FreeGroup(2);;x:=F.1;;y:=F.2;; D:=F/[x^100,y^2,(x*y)^2];;

gap> D_100:=Image(IsomorphismPermGroup(D));;

gap> gens:=GeneratorsOfGroup(D_100);;

gap> R:=ResolutionNormalSeries([gens, gens[1], Identity(D_100)],100);;

gap> TR:=TensorWithIntegers(R);;

gap> Homology(R,99);
[ 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,
100 ]

The Sylow 2-subgroup of the Symmetric group S₂₀has order 2¹⁸. The following commands show that this group G has third homology H₃(G,Z) = (Z₂)⁵⁶+(Z₄)⁷.

gap> G:=SylowSubgroup(SymmetricGroup(20));;

gap> R:=ResolutionNormalSeries(BigStepLCS(G,64),4);;

gap> TR:=TensorWithIntegers(R);;

gap> Homology(R,3);
[ 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, 4, 4, 4, 4, 4, 4, 4 ]

Using the function TwistedTensorProduct() this divide-and-conquer technique can be applied to any group G with a subnormal series G=L₁ > L₂ > ... > L_k. The only requirements are that we have resolutions (with contracting homotopies) for the group L_k and for each quotient L_i/L_i+1. In particular, resolutions for polycyclic groups can be found using the function ResolutionSubnormalSeries(L,n) .

The group G need not be finite. For example, the following commands use the divide-and-conquer technique to compute the integral homology of the free nilpotent group G of class two on four generators. The homology is trivial in dimensions greater than 10, and there is 3-torsion in dimensions 4 and 5. This homology, and also that of the free nilpotent group of class two on five generators, was first calculated in [Larry Lambe, "Cohomology of principal G-bundles over a torus when H^*(BG,R) is polynomial", Bulletin Soc. Math. de Belgium, 38 (1986), 247-264].

(The integral cohomology ring structure for the free nilpotent group of class two on three generators is investigated on a subsequent page. )

The group G is created using the GAP package "nq" by Werner Nickel.

gap> F:=FreeGroup(4);;
gap> G:=NilpotentQuotient(F,2);;

gap> R:=ResolutionNilpotentGroup(G,12);;
gap> TR:=TensorWithIntegers(R);;
gap> for n in [1..11] do
> Print("Homology in dimension ", n," is \n",Homology(TR,n),"\n");
> od;
Homology in dimension 1 is
[ 0, 0, 0, 0 ]
Homology in dimension 2 is
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
Homology in dimension 3 is
[ 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 ]
Homology in dimension 4 is
[ 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 ]
Homology in dimension 5 is
[ 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, 0, 0, 0, 0, 0, 0 ]
Homology in dimension 6 is
[ 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 ]
Homology in dimension 7 is
[ 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 ]
Homology in dimension 8 is
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
Homology in dimension 9 is
[ 0, 0, 0, 0 ]
Homology in dimension 10 is
[ 0 ]
Homology in dimension 11 is
[ ]

By contrast, the following commands show that the integral homology of the free nilpotent group of class three on three generators has 2-torsion in dimension 4.

gap> F:=FreeGroup(3);;
gap> G:=NilpotentQuotient(F,3);;

gap> R:=ResolutionNilpotentGroup(G,5);;
gap> TR:=TensorWithIntegers(R);;
gap> Homology(TR,4); time;
[ 2, 2, 2, 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, 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, 0, 0, 0 ]
46521881

Note the time taken by GAP's Smith Normal Form algorithm to compute the homology in the last example! The 2-torsion can be detected more quickly using the universal coefficient exact sequence for a trivial ZG-module A (where (×) denotes the tensor product over Z).

1 → H_n(G,Z) (×) A → H_n(G,A) → Tor₁(H_n-1(G,Z),A) → 1

The third integral homology of the group G is quickly calculated and seen to be torsion free. So H₄(G,A) is isomorphic to H₄(G,Z) (×) A. The following commands now establish 2-torsion in the fourth integral homology.

gap> F:=FreeGroup(3);;
gap> G:=NilpotentQuotient(F,3);;

gap> R:=ResolutionNilpotentGroup(G,5);;
gap> TRrational:=TensorWithRationals(R);;
gap> TRmod2:=TensorWithIntegersModP(R,2);;

gap> Homology(TRrational,4);time;
171
320702

gap> Homology(TRmod2,4);time;
174
36705

Some of the most interesting groups have no normal subgroup and so the above techniques can't be applied to them. We need additional techniques.

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