Algebraic & Geometric Topology

Equivariant vector bundles over classifying spaces for proper actions

Dieter Degrijse and Ian Leary

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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
Received: 1 May 2015
Revised: 19 May 2016
Accepted: 17 June 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
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

Subjects
Primary: 19L47: Equivariant $K$-theory [See also 55N91, 55P91, 55Q91, 55R91, 55S91]
Secondary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 55N15: $K$-theory [See also 19Lxx] {For algebraic $K$-theory, see 18F25, 19- XX} 55N91: Equivariant homology and cohomology [See also 19L47]

Keywords
equivariant vector bundles classifying spaces for proper actions

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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