


A
GCW
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 pointwise. We denote by O_{G} the category with one object G/H for each finite subgroup H in G, and with maps G/H > G/H' the morphisms of Gsets. A Bredon module is a contravariant functor M:O_{G} > Ab to the category of abelian groups. Standard examples of Bredon modules are:
We denote by H_{n}(X,M) the Bredon homology of a rigid GCW 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 H_{1}(K,B)=0 of
the Quillen complex K(G,p) at the prime p=3 for the symmetric group G=S_{9
}with
coefficients in the Burnside ring B. The simplicial complex
K(G,p) is
the order complex of the poset of nontrivial elementary abelian
psubgroups of G. The Gaction 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); Gchain 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 H_{0}(ESL_{3}(Z),R) = Z^{8}
of a classifying space for proper actions for the special linear group
SL_{3}(Z); the coefficients are in the complex representation
ring R. 

gap>
R:=ContractibleGcomplex("SL(3,Z)s"); Nonfree 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 H_{1}(EG,R) = Z_{2}+Z^{3} 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 ] 

