Algebraic & Geometric Topology

Equivariant vector bundles over classifying spaces for proper actions

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.

Article information

Source
Algebr. Geom. Topol., Volume 17, Number 1 (2017), 131-156.

Dates
Revised: 19 May 2016
Accepted: 17 June 2016
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.agt/1510841311

Digital Object Identifier
doi:10.2140/agt.2017.17.131

Mathematical Reviews number (MathSciNet)
MR3604375

Zentralblatt MATH identifier
1378.19001

Citation

Degrijse, Dieter; Leary, Ian. Equivariant vector bundles over classifying spaces for proper actions. Algebr. Geom. Topol. 17 (2017), no. 1, 131--156. doi:10.2140/agt.2017.17.131. https://projecteuclid.org/euclid.agt/1510841311

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