#### Volume 17, issue 1 (2017)

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Equivariant vector bundles over classifying spaces for proper actions

### Dieter Degrijse and Ian J Leary

Algebraic & Geometric Topology 17 (2017) 131–156
##### Abstract

Let $G$ be an infinite discrete group and let $\underset{¯}{E}G$ be a classifying space for proper actions of $G\phantom{\rule{0.3em}{0ex}}$. Every $G$–equivariant vector bundle over $\underset{¯}{E}G$ gives rise to a compatible collection of representations of the finite subgroups of $G\phantom{\rule{0.3em}{0ex}}$. 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 $\underset{¯}{E}G\phantom{\rule{0.3em}{0ex}}$. 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.

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##### Keywords
equivariant vector bundles, classifying spaces for proper actions
##### Mathematical Subject Classification 2010
Primary: 19L47
Secondary: 20F65, 55N15, 55N91