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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 E ¯G be a classifying space for proper actions of G. Every G–equivariant vector bundle over 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 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 E ¯G. This implies that the Atiyah–Hirzebruch spectral sequence computing the G–equivariant topological K–theory of 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
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