×

zbMATH — the first resource for mathematics

Topological \(K\)-(co)homology of classifying spaces of discrete groups. (English) Zbl 1262.55002
This paper presents a method for computing generalized cohomology and homology of the Borel construction \(EG\times_GX\) on \(X\), where \(G\) is a discrete group and \(X\) is a proper \(G\)-CW-complex. If we focus on the case of a cohomology theory \(\mathcal{H}^*\), then the main result of this paper is Theorem 3.6, in particular its last assertion (iii), which states that under some conditions for appropriate natural numbers \(r^k_p(X)\) there is an exact sequence \[ 0 \to A \to \mathcal{H}^k(G\setminus X) \to \mathcal{H}^k(EG\times_GX) \to B\times \prod_{p \in \mathcal{P}(X)}(\mathbb{Z}^{hat{ \;}}_p)^{r^k_p(X)} \to C \to 0 \] where \(A\), \(B\) and \(C\) are finite abelian groups with \[ A\otimes_{\mathbb{Z}}\mathbb{Z}\left[\frac{1}{\mathcal{P}(X)}\right] =B\otimes_{\mathbb{Z}}\mathbb{Z}\left[\frac{1}{\mathcal{P}(X)}\right] =C\otimes_{\mathbb{Z}}\mathbb{Z}\left[\frac{1}{\mathcal{P}(X)}\right]=0. \] Here \(\mathcal{P}(X)\) denotes the set of primes \(p\) which divides \(|G_x|\) for \(x \in X\). As a particular application of the approach used to get the exact sequence above and its dual, the topological \(K\)-theory of \(BG\) is discussed. Let \(X\) be a finite proper \(G\)-CW-complex satisfying the condition \(\tilde{H}_k(X)=0\), \(k \in \mathbb{Z}\), then it is shown that the above exact sequence can be transformed into an exact sequence \[ 0 \to A \to K^k(G\setminus X) \to K^k(BG) \to B\times \prod_{p \in \mathcal{P}(G)}(\mathbb{Z}^{\hat{ \;}}_p)^{r^k_p(G)} \to C \to 0 \] where \(\mathcal{P}(G)\) is given as the set of primes \(p\) dividing \(|H|\) of some finite subgroup \(H \subset G\) and \(r^k_p(G)\) is given by \[ r^k_p(G)=\sum_{(g) \in \text{con}_p(G)}\sum_{i \in \mathbb{Z}}\text{dim}_{\mathbb{Q}} H^{k+2i}(BC_G(g); \mathbb{Q}), \] \(\text{con}_p(G)\) denoting the set of conjugacy classes of elements of \(p\)-power order in \(G\). Moreover the authors prove universal coefficient theorems for equivariant \(K\)-theory and a cocompletion theorem for equivariant \(K\)-homology in connection with the proof of this \(K\)-theory exact sequence. Finally they give examples of computations which follow directly from assertion (i) of Theorem 3.6, though assertion (iii) also follows from this.
MSC:
55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology
55N15 Topological \(K\)-theory
19L47 Equivariant \(K\)-theory
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] H Abels, A universal proper \(G\)-space, Math. Z. 159 (1978) 143 · Zbl 0388.54027 · doi:10.1007/BF01214487 · eudml:172651
[2] J F Adams, Lectures on generalised cohomology, Springer (1969) 1 · Zbl 0193.51702 · doi:10.1007/BF01580290
[3] A Adem, Characters and \(K\)-theory of discrete groups, Invent. Math. 114 (1993) 489 · Zbl 0804.55002 · doi:10.1007/BF01232678 · eudml:144156 · arxiv:math/9301211
[4] D Anderson, Universal coefficient theorems for \(K\)-theory, mimeographed notes (1969)
[5] M Artin, B Mazur, Etale homotopy, Lecture Notes in Mathematics 100, Springer (1969) · Zbl 0182.26001 · doi:10.1007/BFb0080957
[6] M F Atiyah, I G Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co. (1969) · Zbl 0175.03601 · doi:10.2307/1968941
[7] M F Atiyah, G B Segal, Equivariant \(K\)-theory and completion, J. Differential Geometry 3 (1969) 1 · Zbl 0215.24403 · euclid:jdg/1214428815
[8] P Baum, A Connes, N Higson, Classifying space for proper actions and \(K\)-theory of group \(C^*\)-algebras (editor R S Doran), Contemp. Math. 167, Amer. Math. Soc. (1994) 240 · Zbl 0830.46061 · doi:10.1090/conm/167/1292018
[9] B Blackadar, \(K\)-theory for operator algebras, Mathematical Sciences Research Institute Publications 5, Cambridge Univ. Press (1998) · Zbl 0913.46054
[10] M Boekstedt, Universal coefficient theorems for equivariant \(K\)- and \(KO\)-theory, preprint (1981/82) · Zbl 0478.55002
[11] A Borel, J P Serre, Corners and arithmetic groups, Comment. Math. Helv. 48 (1973) 436 · Zbl 0274.22011 · doi:10.1007/BF02566134 · eudml:139559
[12] G E Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics 46, Academic Press (1972) · Zbl 0246.57017 · doi:10.1090/S0002-9904-1939-07038-9
[13] K S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer (1982) · Zbl 0584.20036
[14] T tom Dieck, Transformation groups, De Gruyter Studies in Mathematics 8, Walter de Gruyter & Co. (1987) · Zbl 0611.57002 · doi:10.1515/9783110858372.312
[15] P Green, The local structure of twisted covariance algebras, Acta Math. 140 (1978) 191 · Zbl 0407.46053 · doi:10.1007/BF02392308
[16] J P C Greenlees, \(K\)-homology of universal spaces and local cohomology of the representation ring, Topology 32 (1993) 295 · Zbl 0779.55005 · doi:10.1016/0040-9383(93)90021-M
[17] A Grothendieck, Sur quelques points d’algèbre homologique, Tôhoku Math. J. 9 (1957) 119 · Zbl 0118.26104 · euclid:dmj/1077491411
[18] I J Leary, B E A Nucinkis, Every CW-complex is a classifying space for proper bundles, Topology 40 (2001) 539 · Zbl 0983.55010 · doi:10.1016/S0040-9383(99)00073-7
[19] W Lück, Transformation groups and algebraic \(K\)-theory, Lecture Notes in Mathematics 1408, Springer (1989) · Zbl 0679.57022 · doi:10.1007/BFb0083681
[20] W Lück, Survey on classifying spaces for families of subgroups (editors L Bartholdi, T Ceccherini-Silberstein, T Smirnova-Nagnibeda, A Zuk), Progr. Math. 248, Birkhäuser (2005) 269 · Zbl 1117.55013 · doi:10.1007/3-7643-7447-0_7
[21] W Lück, Rational computations of the topological \(K\)-theory of classifying spaces of discrete groups, J. Reine Angew. Math. 611 (2007) 163 · Zbl 1144.55007 · doi:10.1515/CRELLE.2007.078
[22] W Lück, B Oliver, Chern characters for the equivariant \(K\)-theory of proper \(G\)-CW-complexes (editors J Aguadé, C Broto, C Casacuberta), Progr. Math. 196, Birkhäuser (2001) 217 · Zbl 0988.55005
[23] W Lück, B Oliver, The completion theorem in \(K\)-theory for proper actions of a discrete group, Topology 40 (2001) 585 · Zbl 0981.55002 · doi:10.1016/S0040-9383(99)00077-4
[24] W Lück, R Stamm, Computations of \(K\)- and \(L\)-theory of cocompact planar groups, \(K\)-Theory 21 (2000) 249 · Zbl 0979.19003 · doi:10.1023/A:1026539221644
[25] W Lück, M Weiermann, On the classifying space of the family of virtually cyclic subgroups, Pure Appl. Math. Q. 8 (2012) 497 · Zbl 1258.55011 · doi:10.4310/PAMQ.2012.v8.n2.a6
[26] R C Lyndon, P E Schupp, Combinatorial group theory, Ergeb. Math. Grenzgeb. 89, Springer (1977) · Zbl 0368.20023
[27] I Madsen, Geometric equivariant bordism and \(K\)-theory, Topology 25 (1986) 217 · Zbl 0595.55006 · doi:10.1016/0040-9383(86)90040-6
[28] D Meintrup, On the type of the universal space for a family of subgroupsat Münster” (editor C Deninger), 3 26, Univ. Münster (2000) 60 · Zbl 0948.22005
[29] D Meintrup, T Schick, A model for the universal space for proper actions of a hyperbolic group, New York J. Math. 8 (2002) 1 · Zbl 0990.20027 · emis:journals/NYJM/j/2002/8-1nf.htm · eudml:121926 · nyjm.albany.edu:8000
[30] G Mislin, Classifying spaces for proper actions of mapping class groups, Münster J. Math. 3 (2010) 263 · Zbl 1223.57017
[31] N C Phillips, Equivariant \(K\)-theory for proper actions, Pitman Research Notes in Mathematics Series 178, Longman Scientific & Technical (1989) · Zbl 0684.55018
[32] J Rosenberg, C Schochet, The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized \(K\)-functor, Duke Math. J. 55 (1987) 431 · Zbl 0644.46051 · doi:10.1215/S0012-7094-87-05524-4
[33] J P Serre, Arithmetic groups (editor C T C Wall), London Math. Soc. Lecture Note Ser. 36, Cambridge Univ. Press (1979) 105 · Zbl 0432.20042
[34] C Soulé, The cohomology of \(\mathrm{SL}_3(\mathbbZ)\), Topology 17 (1978) 1 · Zbl 0382.57026 · doi:10.1016/0040-9383(78)90009-5
[35] R M Switzer, Algebraic topology-homotopy and homology, Grundl. Math. Wissen. 212, Springer (1975) · Zbl 0305.55001
[36] M Tezuka, N Yagita, Complex \(K\)-theory of \(B\mathrm{SL}_3(\mathbbZ)\), \(K\)-Theory 6 (1992) 87 · Zbl 0770.55007 · doi:10.1007/BF00961336
[37] C A Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge Univ. Press (1994) · Zbl 0797.18001
[38] G W Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics 61, Springer (1978) · Zbl 0406.55001
[39] N Yoneda, On Ext and exact sequences, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960) 507 · Zbl 0163.26902
[40] Z i Yosimura, Universal coefficient sequences for cohomology theories of CW-spectra, Osaka J. Math. 12 (1975) 305 · Zbl 0309.55008 · euclid:ojm/1200757858
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.