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.
The following functions for computing Bredon homology are joint work with Bui Anh Tuan.
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
coefficients in the Burnside ring B. The simplicial complex
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
Simplicial complex of dimension 2.
G-chain complex in characteristic 0 for Sym( [ 1 .. 9 ] ) .
Chain complex of length 2 in characteristic 0 .
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
Non-free resolution in characteristic 0 for <matrix group> .
Chain complex of length 3 in characteristic 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.|
[ 2, 0, 0, 0 ]