Algebraic & Geometric Topology

Equivariant vector bundles over classifying spaces for proper actions

Dieter Degrijse and Ian Leary

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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

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

equivariant vector bundles classifying spaces for proper actions


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.

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  • M,F Atiyah, Characters and cohomology of finite groups, Inst. Hautes Études Sci. Publ. Math. 9 (1961) 23–64
  • G,E Bredon, Equivariant cohomology theories, Lecture Notes in Mathematics 34, Springer, Berlin (1967)
  • M,R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grundl. Math. Wissen. 319, Springer, Berlin (1999)
  • J,F Davis, W Lück, Spaces over a category and assembly maps in isomorphism conjectures in $K$– and $L$–theory, $K$–Theory 15 (1998) 201–252
  • M,W Davis, The geometry and topology of Coxeter groups, London Mathematical Society Monographs Series 32, Princeton University Press (2008)
  • T tom Dieck, Transformation groups, de Gruyter Studies in Mathematics 8, de Gruyter, Berlin (1987)
  • R,A Horn, C,R Johnson, Matrix analysis, 2nd edition, Cambridge University Press (2013)
  • S Jackowski, Families of subgroups and completion, J. Pure Appl. Algebra 37 (1985) 167–179
  • I,J Leary, A metric Kan–Thurston theorem, J. Topol. 6 (2013) 251–284
  • W Lück, The Burnside ring and equivariant stable cohomotopy for infinite groups, Pure Appl. Math. Q. 1 (2005) 479–541
  • W Lück, Equivariant cohomological Chern characters, Internat. J. Algebra Comput. 15 (2005) 1025–1052
  • W Lück, Survey on classifying spaces for families of subgroups, from “Infinite groups: geometric, combinatorial and dynamical aspects” (L Bartholdi, T Ceccherini-Silberstein, T Smirnova-Nagnibeda, A Zuk, editors), Progr. Math. 248, Birkhäuser, Basel (2005) 269–322
  • W Lück, B Oliver, Chern characters for the equivariant $K$–theory of proper $G$–CW–complexes, from “Cohomological methods in homotopy theory” (J Aguadé, C Broto, C Casacuberta, editors), Progr. Math. 196, Birkhäuser, Basel (2001) 217–247
  • W Lück, B Oliver, The completion theorem in $K$–theory for proper actions of a discrete group, Topology 40 (2001) 585–616
  • J,P May, Equivariant homotopy and cohomology theory, from “Symposium on algebraic topology in honor of José Adem” (S Gitler, editor), Contemp. Math. 12, Amer. Math. Soc., Providence, RI (1982) 209–217
  • J,R Munkres, Elements of algebraic topology, Addison-Wesley, Menlo Park, CA (1984)
  • R,J Sánchez-García, Equivariant $K$–homology for some Coxeter groups, J. Lond. Math. Soc. 75 (2007) 773–790
  • G Segal, Equivariant $K$–theory, Inst. Hautes Études Sci. Publ. Math. 34 (1968) 129–151
  • J-P Serre, Linear representations of finite groups, Graduate Texts in Mathematics 42, Springer (1977)