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 xG 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 <>