P-Toral Approximations Compute Bredon Homology

Abstract

Abstract We study Bredon homology approximations for spaces with an action of a compact Lie group G. We show that if M is a coMackey functor satisfying mild p-locality conditions, then Bredon homology of a G-space X with coefficients in M is determined by fixed points of p-toral subgroups of G acting on X. As an application we prove a vanishing result for the Bredon homology of the complex ${{\mathcal{L}}}_{n}$ of direct-sum decompositions of ${{\mathbb{C}}}^{n}$.