Nullification functors and the homotopy type of the classifying space for proper bundles

Abstract

Let $G$ be a discrete group for which the classifying space for proper $G$–actions is finite-dimensional. We find a space $W$ such that for any such $G$, the classifying space $\underset{¯}{\mathrm{\text{ B}}}G$ for proper $G$–bundles has the homotopy type of the $W$–nullification of $\mathrm{\text{ B}}G$. We use this to deduce some results concerning $\underset{¯}{\mathrm{\text{ B}}}G$ and in some cases where there is a good model for $\underset{¯}{\mathrm{\text{ B}}}G$ we obtain information about the $\mathrm{\text{ B}}\mathbb{Z}∕p$–nullification of $\mathrm{\text{ B}}G$.