Equivariant vector bundles over classifying spaces for proper actions

Abstract

Let $G$ be an infinite discrete group and let $\underset{¯}{E}G$ be a classifying space for proper actions of $G$. Every $G$–equivariant vector bundle over $\underset{¯}{E}G$ gives rise to a compatible collection of representations of the finite subgroups of $G$. We give the first examples of groups $G$ with a cocompact classifying space for proper actions $\underset{¯}{E}G$ admitting a compatible collection of representations of the finite subgroups of $G$ that does not come from a $G$–equivariant (virtual) vector bundle over $\underset{¯}{E}G$. This implies that the Atiyah–Hirzebruch spectral sequence computing the $G$–equivariant topological $K$–theory of $\underset{¯}{E}G$ has nonzero differentials. On the other hand, we show that for right-angled Coxeter groups this spectral sequence always collapses at the second page and compute the $K$–theory of the classifying space of a right-angled Coxeter group.