P-Toral Approximations Compute Bredon Homology

TL;DR

This paper introduces p-toral approximations to compute Bredon homology for G-spaces, showing that fixed points of p-toral subgroups determine the homology under certain conditions, with applications to complex decompositions.

Contribution

It establishes a method to compute Bredon homology via p-toral subgroup fixed points and proves a vanishing result for complex L_n homology.

Findings

Bredon homology is determined by p-toral subgroup fixed points.

A vanishing theorem for the Bredon homology of complex L_n.

Conditions under which p-toral approximations are effective.

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 L_n of direct-sum decompositions of complex n-space.