On the K-theory of Lie groups (original) (raw)

Torsion in cohomology of compact Lie groups and Chow rings of reductive algebraic groups

Inventiones Mathematicae, 1985

Dedicated to E.B. Dynkin on his 60th birthday The purpose of the present note is to give a general procedure for calculating the singular cohomology H*(K, IFp) with coefficients in an arbitrary finite field IFv of a connected compact Lie group K. Simultaneously, we calculate the Chow ring A(G, IFp)..=A(G)| p of the corresponding complex algebraic group G, and the degrees of the basic "generalized invariants" of the Weyl group W in arbitrary characteristic p. A mysterious connection between the degrees of generators of A(G, IFp) and the kernel of the principal nilpotent element in characteristic p is pointed out. The computation of real cohomology of compact Lie groups started by E. Cartan in 1929 was completed for classical groups by R. Brauer and by L.

On the cohomology of the classical linear groups

Illinois Journal of Mathematics, 1976

In this paper we use the methods of [1] to partially compute the cohomology of the classical groups with coefficients in the finite field with q elements, Fq. Here q is a power of an oddprimep. Cohomology is the usual group cohomology of Eilenberg-MacLane [-2] and coefficients are taken in Z, the integers mod l, where is a prime different from p. Inherent in this method is the equivalence between the group cohomology of G, H*(G), and the singular cohomology of BG, H*(BG), where BG is a classifying space for G (see for example [3, pp. 185-186]). In this paper we will freely interchange these two concepts. The approach as in [1] is to tie the cohomology of BG to the cohomology of B U, where U is the infinite unitary group. This is done by the use of a virtual complex representation induced from the natural modular representation of G on F [4, Theorem 1-]. Strong use is made of the classical Lie theory associated to these groups by Chevalley [5] (e.g., the action of a Weyl group on diagonal subgroups of G is critical for the analysis). In one form the main theorem says that the cohomology of G is generated by Chern classes (see [6, Appendix]). As in [-1] we must pass to a certain subfield, k, of the algebraic closure of Fq in order to complete the computations. Let T denote the diagonal subgroup of G [7, chapter 7.-] and W the Weyl group of G. Another form of the main theorem says that H*(G)-H*(T) w, the fixed subring of H*(T) under the induced action of W. This theorem was proved in [ ] for GL,(k 2) and O,(kl), the general linear and orthogonal groups. In this paper we extend the results to the other classical groups SL,(kl), the special linear groups, SPzm(k) the symplectic groups and if q is an even power ofp U,(k), the unitary groups. No attempt is made to complete the results in Fq itself as is done for GL,(Fq) in [8]. 1. Definitions Let p be any odd prime and q pS where s is a positive integer. Fq will stand for the finite field with q elements and GL,(Fq) will be the general linear group over Fq (i.e., elements of GL,(Fq) are the n x n matrices with coefficients in Fq whose determinant is nonzero). We will consider a number of other classical linear groups and view them as subgroups of GL,(Fq). The easiest to define is the subgroup of elements whose determinant is 1. This subgroup is denoted by SL,(Fq), the special linear 9roup.

Equivariant geometric K-homology for compact Lie group actions

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 2010

Let G be a compact Lie-group, X a compact G-CW-complex. We define equivariant geometric K-homology groups K G * (X), using an obvious equivariant version of the (M, E, f )-picture of Baum-Douglas for K-homology. We define explicit natural transformations to and from equivariant K-homology defined via KK-theory (the "official" equivariant K-homology groups) and show that these are isomorphisms.

Chern characters for the equivariant K-theory of proper G-CW-complexes

Cohomological Methods in Homotopy Theory, 2001

In an earlier paper [10], we showed that for any discrete group G, equivariant K-theory for finite proper G-CW-complexes can be defined using equivariant vector bundles. This was then used to prove a version of the Atiyah-Segal completion theorem in this situation. In this paper, we continue to restrict attention to actions of discrete groups, and begin by constructing an appropriate classifying space which allows us to define K * G (X) for an arbitrary proper G-complex X. We then construct rational-valued equivariant Chern characters for such spaces, and use them to prove some more general versions of completion theorems. In fact, we construct two different types of equivariant Chern character, both of which involve Bredon cohomology with coefficients in the system G/H → R(H). The first, ch

Equivariant K-theory of compact Lie groups with involution

Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology, 2014

For a compact simply connected simple Lie group G with an involution α, we compute the G ⋊ ℤ/2-equivariant K-theory of G where G acts by conjugation and ℤ/2 acts either by α or by g ↦ α(g)−1. We also give a representation-theoretic interpretation of those groups, as well as of KG(G).

A geometric description of equivariant K-homology for proper actions

2009

Let G be a discrete group and let X be a G-finite, proper G-CW-complex. We prove that Kasparov's equivariant K-homology groups KK G * (C 0 (X), C) are isomorphic to the geometric equivariant K-homology groups of X that are obtained by making the geometric K-homology theory of Baum and Douglas equivariant in the natural way. This reconciles the original and current formulations of the Baum-Connes conjecture for discrete groups.

Some properties of Lubin-Tate cohomology for classifying spaces of finite groups

Arxiv preprint arXiv:1005.1662, 2010

We consider brave new cochain extensions F (BG+, R) −→ F (EG+, R), where R is either a Lubin-Tate spectrum En or the related 2-periodic Morava K-theory Kn, and G is a finite group. When R is an Eilenberg-Mac Lane spectrum, in some good cases such an extension is a G-Galois extension in the sense of John Rognes, but not always faithful. We prove that for En and Kn these extensions are always faithful in the Kn local category. However, for a cyclic p-group Cpr , the cochain extension F (BCpr + , En) −→ F (ECpr + , En) is not a Galois extension because it ramifies. As a consequence, it follows that the En-theory Eilenberg-Moore spectral sequence for G and BG does not always converge to its expected target.