Equivariant vector bundles over classifying spaces for proper actions

Degrijse, Dieter; Leary, Ian

doi:10.2140/agt.2017.17.131

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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

DOI: 10.2140/agt.2017.17.131

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–Hirzebruch spectral sequence computing the $G$ –equivariant topological $K$ –theory of $\underline{E} G$ 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.

Keywords

equivariant vector bundles, classifying spaces for proper actions

Mathematical Subject Classification 2010

Primary: 19L47

Secondary: 20F65, 55N15, 55N91

References

Publication

Received: 1 May 2015

Revised: 19 May 2016

Accepted: 17 June 2016

Published: 26 January 2017

Authors

Dieter Degrijse
Department of Mathematical Sciences University of Copenhagen Universitetsparken 2100 Copenhagen Denmark	School of Mathematics, Statistics & Applied Mathematics NUI Galway University Road Galway Ireland

Ian J Leary
School of Mathematical Sciences University of Southampton Southhampton SO17 1BJ United Kingdom

Volume 17, issue 1 (2017)