\title{Homology decompositions for classifying spaces of compact Lie groups}
\author{Alexei Strounine}
\address{Mathematics Department, University of Notre Dame, Notre Dame, IN 46556}
\subjclass{Primary 55R35; Secondary 55R40}

\begin{abstract}
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. In \cite{JMO} Jackowski, McClure and Oliver 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.
\end{abstract}