About HAP: Bredon Homology

A G-CW complex X is a CW space with an action of a group G that induces a permutation of cells. The space is said to be rigid if any element of G that stabilizes a cell stabilizes it point-wise.

We denote by OG the category with one object G/H for each finite subgroup H in G, and with maps G/H --> G/H' the morphisms of G-sets.

A Bredon module is a contravariant functor M:OG ---> Ab to the category of abelian groups.

Standard examples of Bredon modules are:
  • the contravariant functor M=B that sends an object G/H to the free abelian group BS(H) with isomorphism types of transitive H-sets as basis.
  • the contravariant functor M=R  that sends an object G/H  to the vector space RC(H) of complex representations of the finite group H.

We denote by Hn(X,M) the Bredon homology of a rigid G-CW space with coefficients in a Bredon module M.

The following functions for computing Bredon homology are joint work with Bui Anh Tuan.
The following commands compute the Bredon homology H1(K,B)=0 of the Quillen complex K(G,p) at the prime p=3 for the symmetric group G=S9 with coefficients in the Burnside ring B.  The simplicial complex K(G,p) is the order complex of the poset of non-trivial elementary abelian p-subgroups of G. The G-action on K is induced by congugation and is rigid.
gap> G:=SymmetricGroup(9);;
gap> K:=QuillenComplex(G,3);
Simplicial complex of dimension 2.
gap> R:=GChainComplex(K,G);
G-chain complex in characteristic 0 for Sym( [ 1 .. 9 ] ) .
gap> C:=TensorWithBurnsideRing(R);
Chain complex of length 2 in characteristic 0 .
gap> Homology(C,1);
[  ]
The following commands compute the the Bredon homology H0(ESL3(Z),R) = Z8 of a classifying space for proper actions for the special linear group SL3(Z); the coefficients are in the complex representation ring R.
gap> R:=ContractibleGcomplex("SL(3,Z)s");
Non-free resolution in characteristic 0 for <matrix group> .

gap> D:=TensorWithComplexRepresentationRing(R);
Chain complex of length 3 in characteristic 0 .

gap> Homology(D,0);
[ 0, 0, 0, 0, 0, 0, 0, 0 ]
The following commands compute the the Bredon homology H1(EG,R) = Z2+Z3 of a classifying space for proper actions for the crystallographic group G=SpaceGroup(3,32); the coefficients are in the Burnside ring B.
gap> G:=SpaceGroup(3,32);;
gap> gens:=GeneratorsOfGroup(G);;
gap> bas:=CrystGFullBasis(G);;
gap> R:=CrystGcomplex(gens,bas,0);;
gap> D:=TensorWithBurnsideRing(R);;
gap> Homology(D,1);
[ 2, 0, 0, 0 ]
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