### An isomorphism between Bredon and Quinn homology via homotopy colimits, by Robert N. Talbert

Consider the stratified system of fibrations *p* from the Borel
construction *EG* x*G* *X* to the orbit
space *X/G* where *G* is a discrete group acting cellularly on the
simplicial complex *X*. In this paper, we show an isomorphism between
the Bredon homology of *X* with a certain coefficient system and the
Quinn homology of *X/G* with coefficients in the spectral sheaf
*S(p)* where *S* is any homotopy-invariant functor from spaces to
spectra (e.g., the K-theory spectrum functor).

The isomorphism comes from first expressing the Quinn homology spectrum as a
homotopy colimit of a certain spectrum-valued functor. We then show that
Bredon homology may be expressed as the homology of a category when *G*
is discrete and acting cellularly on a simplicial complex, and then this
homology may be computed as the homotopy groups of a homotopy colimit of a
spectrum-valued functor. The homology isomorphism is obtained on the homotopy
colimit level. As a by-product, we get a completely algebraic description of
Quinn homology with stratified abelian group coefficients via the homology of
categories.

Robert N. Talbert <talbertr@bethel-in.edu>