Equivariant vector bundles over classifying spaces for proper actions

Degrijse, Dieter¹ ; Leary, Ian²

¹Department of Mathematical Sciences, University of Copenhagen, Universitetsparken, 2100 Copenhagen, Denmark, School of Mathematics, Statistics & Applied Mathematics, NUI Galway, University Road, Galway, Ireland
²School of Mathematical Sciences, University of Southampton, Southhampton, SO17 1BJ, United Kingdom

Algebraic and Geometric Topology, Tome 17 (2017) no. 1, pp. 131-156

Résumé

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.

Détail

DOI : 10.2140/agt.2017.17.131

Classification : 19L47, 20F65, 55N15, 55N91
Keywords: equivariant vector bundles, classifying spaces for proper actions

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