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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:
We denote by Hn(X,M) the Bredon homology of a rigid G-CW space with coefficients in a Bredon module M. |
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The
following commands compute the Bredon homology H1(K,B)=0 of
the Quillen complex K at the prime p=3 for the symmetric group S9 with
coefficients in the Burnsidee ring B. The simplicial complex K is
the order complex of the poset of non-trivial elementary abelian
subgroups of G. The G-action on K is induced by congugation. |
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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); [ ] |
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