Algebraic & Geometric Topology

Topological $K$–(co)homology of classifying spaces of discrete groups

Michael Joachim and Wolfgang Lück

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Let G be a discrete group. We give methods to compute, for a generalized (co)homology theory, its values on the Borel construction EG×GX of a proper G–CW–complex X satisfying certain finiteness conditions. In particular we give formulas computing the topological K–(co)homology K(BG) and K(BG) up to finite abelian torsion groups. They apply for instance to arithmetic groups, word hyperbolic groups, mapping class groups and discrete cocompact subgroups of almost connected Lie groups. For finite groups G these formulas are sharp. The main new tools we use for the K–theory calculation are a Cocompletion Theorem and Equivariant Universal Coefficient Theorems which are of independent interest. In the case where G is a finite group these theorems reduce to well-known results of Greenlees and Bökstedt.

Article information

Algebr. Geom. Topol., Volume 13, Number 1 (2013), 1-34.

Received: 25 January 2012
Accepted: 14 August 2012
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55N20: Generalized (extraordinary) homology and cohomology theories
Secondary: 55N15: $K$-theory [See also 19Lxx] {For algebraic $K$-theory, see 18F25, 19- XX} 19L47: Equivariant $K$-theory [See also 55N91, 55P91, 55Q91, 55R91, 55S91]

Classifying spaces Topological $K$–theory


Joachim, Michael; Lück, Wolfgang. Topological $K$–(co)homology of classifying spaces of discrete groups. Algebr. Geom. Topol. 13 (2013), no. 1, 1--34. doi:10.2140/agt.2013.13.1.

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  • H Abels, A universal proper $G$–space, Math. Z. 159 (1978) 143–158
  • J F Adams, Lectures on generalised cohomology, from: “Category theory, homology theory and their applications III”, Springer, Berlin (1969) 1–138
  • A Adem, Characters and $K$–theory of discrete groups, Invent. Math. 114 (1993) 489–514
  • D Anderson, Universal coefficient theorems for $K$–theory, mimeographed notes, University of California, Berkeley (1969)
  • M Artin, B Mazur, Etale homotopy, Lecture Notes in Mathematics 100, Springer, Berlin (1969)
  • M F Atiyah, I G Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, MA (1969)
  • M F Atiyah, G B Segal, Equivariant $K$–theory and completion, J. Differential Geometry 3 (1969) 1–18
  • P Baum, A Connes, N Higson, Classifying space for proper actions and $K$–theory of group $C^{*}$–algebras, from: “$C^{*}$–algebras: 1943–1993”, (R S Doran, editor), Contemp. Math. 167, Amer. Math. Soc. (1994) 240–291
  • B Blackadar, $K$–theory for operator algebras, 2nd edition, Mathematical Sciences Research Institute Publications 5, Cambridge Univ. Press (1998)
  • M Boekstedt, Universal coefficient theorems for equivariant $K$– and $KO$–theory, preprint 7, Univ. Aarhus (1981/82)
  • A Borel, J-P Serre, Corners and arithmetic groups, Comment. Math. Helv. 48 (1973) 436–491
  • G E Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics 46, Academic Press, New York (1972)
  • K S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer, New York (1982)
  • T tom Dieck, Transformation groups, De Gruyter Studies in Mathematics 8, Walter de Gruyter & Co., Berlin (1987)
  • P Green, The local structure of twisted covariance algebras, Acta Math. 140 (1978) 191–250
  • J P C Greenlees, $K$–homology of universal spaces and local cohomology of the representation ring, Topology 32 (1993) 295–308
  • A Grothendieck, Sur quelques points d'algèbre homologique, Tôhoku Math. J. 9 (1957) 119–221
  • I J Leary, B E A Nucinkis, Every CW–complex is a classifying space for proper bundles, Topology 40 (2001) 539–550
  • W Lück, Transformation groups and algebraic $K$–theory, Lecture Notes in Mathematics 1408, Springer, Berlin (1989)
  • 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, Rational computations of the topological $K$–theory of classifying spaces of discrete groups, J. Reine Angew. Math. 611 (2007) 163–187
  • 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
  • W Lück, R Stamm, Computations of $K$– and $L$–theory of cocompact planar groups, $K$–Theory 21 (2000) 249–292
  • W Lück, M Weiermann, On the classifying space of the family of virtually cyclic subgroups, Pure Appl. Math. Q. 8 (2012) 497–555
  • R C Lyndon, P E Schupp, Combinatorial group theory, Ergeb. Math. Grenzgeb. 89, Springer, Berlin (1977)
  • I Madsen, Geometric equivariant bordism and $K$–theory, Topology 25 (1986) 217–227
  • D Meintrup, On the type of the universal space for a family of subgroups, from: “Schriftenreihe des Mathematischen Instituts der Universität Münster”, (C Deninger, editor), 3 26, Univ. Münster (2000) 60
  • D Meintrup, T Schick, A model for the universal space for proper actions of a hyperbolic group, New York J. Math. 8 (2002) 1–7
  • G Mislin, Classifying spaces for proper actions of mapping class groups, Münster J. Math. 3 (2010) 263–272
  • N C Phillips, Equivariant $K$–theory for proper actions, Pitman Research Notes in Mathematics Series 178, Longman Scientific & Technical, Harlow (1989)
  • J Rosenberg, C Schochet, The Künneth theorem and the universal coefficient theorem for Kasparov's generalized $K$–functor, Duke Math. J. 55 (1987) 431–474
  • J-P Serre, Arithmetic groups, from: “Homological group theory”, (C T C Wall, editor), London Math. Soc. Lecture Note Ser. 36, Cambridge Univ. Press (1979) 105–136
  • C Soulé, The cohomology of $\mathrm{SL}_{3}(\mathbb{Z})$, Topology 17 (1978) 1–22
  • R M Switzer, Algebraic topology–-homotopy and homology, Grundl. Math. Wissen. 212, Springer, New York (1975)
  • M Tezuka, N Yagita, Complex $K$–theory of $B\mathrm{SL}_3(\mathbb{Z})$, $K$–Theory 6 (1992) 87–95
  • C A Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge Univ. Press (1994)
  • G W Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics 61, Springer, New York (1978)
  • N Yoneda, On Ext and exact sequences, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960) 507–576
  • Z-i Yosimura, Universal coefficient sequences for cohomology theories of CW–spectra, Osaka J. Math. 12 (1975) 305–323