Centralizers of elementary abelian p-subgroups, the Borel construction of the singular locus and applications to the cohomology of discrete groups

Abstract

Let G be either a compact Lie group or a discrete group and X a space with a continuous G action. Assume the action has finite stabilizers if G is discrete. For a fixed prime p let Xs be the p singular locus of X, i.e. the set of all points in X which are fixed by some element of order p. In this paper we study the Borel construction EG×G Xs (EG denoting as usual the total space of the universal principal G bundle) by relating it to the Borel constructions EG×CG(E)X where E runs through the non-trivial elementary abelian p subgroups of G, CG(E) denotes the centralizer of E in G and X E is the E fixed point set of X.