Homology decompositions for classifying spaces of compact Lie groups

Let p be a prime number and G be a compact Lie group. A homology decomposition for the classifying space BG is a way of building BG up to mod p homology as a homotopy colimit of classifying spaces of subgroups of G. In this paper we develop techniques for constructing such homology decompositions. Jackowski, McClure and Oliver (Homotopy classification of self-maps of BG via G-actions, Ann. of Math. 135 (1992), 183-270) construct a homology decomposition of BG by classifying spaces of p-stubborn subgroups of G. Their decomposition is based on the existence of a finite-dimensional mod p acyclic G-CW-complex with restricted set of orbit types. We apply our techniques to give a parallel proof of the p-stubborn decomposition of BG which does not use this geometric construction.