# zbMATH — the first resource for mathematics

On the $$K$$-theory of the classifying space of a discrete group. (English) Zbl 0761.55012
Math. Ann. 292, No. 2, 319-327 (1992); erratum ibid. 293, No. 2, 385-386 (1992).
The author establishes a decomposition for the equivariant $$K$$-theory of a discrete group $$\Gamma$$ with suitable assumptions. Let $$X$$ be an admissible $$\Gamma$$-complex (i.e.: a finite-dimensional contractible $$\Gamma$$-complex such that the finite subgroups are exactly the isotropy subgroups and the fixed-point set $$X^ H$$ are contractible) and let $$\Gamma'$$ be a normal torsion-free subgroup in $$\Gamma$$ with finite factor group $$G$$, where $$\Gamma$$ have finite virtual cohomological dimension. Then $$\Gamma'$$ acts freely on $$X$$ and we have, by the completion theorem, an isomorphism: $$K^*_ G(X/\Gamma')^ \wedge\cong K^*(B\Gamma)$$.
Using a recent result of Atiyah and Segal (expressing $$K^*_ G(X)\otimes\mathbb{C}$$ as a sum of $$K^*(Y^{\langle g\rangle}/C(g))\otimes\mathbb{C}$$, as $$g$$ ranges over conjugacy classes of elements in $$G)$$, the author indicates that the $$K$$-theory of $$B\Gamma$$ can be calculated given enough information about its elements of finite order and their centralizers — see formula (3.1).
Specific examples presented at the end of this note include amalgamated products as well as certain arithmetic groups.

##### MSC:
 55R35 Classifying spaces of groups and $$H$$-spaces in algebraic topology
##### Keywords:
classifying space; equivariant $$K$$-theory; discrete group
Full Text:
##### References:
 [1] Ash, A.: Farrell cohomology of GL n (?). Isr. J. Math. (to appear) · Zbl 0693.20041 [2] Atiyah, M.F., Segal, G.B.: EquivariantK-theory and completion. J. Differ. Geom.3, 1-18 (1969) · Zbl 0215.24403 [3] Atiyah, M.F., Segal, G.B.: Unpublished [4] Brown K.: Cohomology of groups. (Grad. Texts Math., vol. 87) Berlin Heidelberg New York: Springer 1982 · Zbl 0584.20036 [5] Brown, K.: Euler characteristics of discrete groups andG-spaces. Invent. Math.27, 229-264 (1974) · Zbl 0294.20047 [6] Brown, K.: Complete Euler characteristics and fixed-point theory. J. Pure Appl. Algebra24, 103-121 (1982) · Zbl 0493.20033 [7] Brown, K.: High-dimensional cohomology of discrete groups. Proc. Natl. Acad. Sci. USA73 (No. 6), 1795-1797 (1976) · Zbl 0339.20013 [8] Eilenberg, S., Ganea, T.: On the Lusternik-Schnirelmann category of abstract groups. Ann. Math.65, 517-518 (1957) · Zbl 0079.25401 [9] Hopkins, M.: Characters and elliptic cohomology. In: Salamon, Steer, Sutherland (eds.) Advances in homotopy theory. (Lond. Math. Soc. Lect. Note Ser., vol. 139), Cambridge New York London: Cambridge University Press 1989 [10] Hirzebruch, F., H?fer, T.: On the Euler number of an orbifold. Math. Ann.286, 255-260 (1990) · Zbl 0679.14006 [11] Segal, G.B.: EquivariantK-theory. Publ. Math., Inst. Hautes ?tud. Sci.34 (1968) [12] Serre, J-P.: Cohomologie des groupes discretes. Ann. Math. Stud.70, 77-169 (1971) [13] Soul?, C.: The cohomology of SL3?. Topology17, 1-22 (1978) · Zbl 0382.57026 [14] Tom Dieck, T.: Transformation groups and representation theory. (Lect. Notes Math., vol. 766) Berlin Heidelberg New York: Springer 1979 · Zbl 0445.57023
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.