Equivariant vector bundles over classifying spaces for proper actions

TL;DR

This paper explores the relationship between equivariant vector bundles and representations over classifying spaces for proper actions, providing new examples and computations in equivariant K-theory, especially for right angled Coxeter groups.

Contribution

It presents the first examples of groups with compatible finite subgroup representations not arising from equivariant vector bundles, and analyzes the spectral sequence behavior in these cases.

Findings

Existence of groups with compatible representations not from vector bundles

Spectral sequence for equivariant K-theory can have non-zero differentials

Computed K-theory for classifying spaces of right angled Coxeter groups

Abstract

Let $G$ be an infinite discrete group and let $\underline{E}G$ be a classifying space for proper actions of $G$. Every $G$-equivariant vector bundle over $\underline{E}G$ 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 $\underline{E}G$ admitting a compatible collection of representations of the finite subgroups of $G$ that does not come from a $G$-equivariant (virtual) vector bundle over $\underline{E}G$. This implies that the Atiyah-Hirzeburch spectral sequence computing the $G$-equivariant topological $K$-theory of $\underline{E}G$ has non-zero differentials. On the other hand, we show that for right angled Coxeter groups this spectral sequence always collapes at the second page and compute the $K$-theory of the classifying space of a right angled Coxeter group.