Going-down functors, the K�nneth formula, and the Baum-Connes conjecture

Abstract

We study the connection between the Baum-Connes conjecture for a locally compact group G with coefficient A and the Künneth formula for the K-theory of tensor products by the corresponding crossed product A r G. The main tool for this is obtained by an application of a general reduction procedure which allows us to analyze certain functors connected to the topological K-theory of a group in terms of their restrictions to compact subgroups. We also discuss several other interesting applications of this method, including a general extension result for the Baum-Connes conjecture.

The Baum–Connes conjecture via localisation of categories

“…Such permanence results have been investigated by several authors. There is a series of papers by Chabert et al [10][11][12]15,16]. Both authors of this article have been quite familiar with their work, and it has greatly influenced this article.…”

“…Since G L r always satisfies the UCT, the strong Baum-Connes conjecture with trivial coefficients holds if and only if the usual Baum-Connes conjecture holds and C * r (G) satisfies the UCT. This is known to be the case for almost connected groups and linear algebraic groups over p-adic number fields, see [14,16]. This article is the first step in a programme to extend the Baum-Connes conjecture to quantum group crossed products.…”

Euler characteristics and Gysin sequences for group actions on boundaries

“…SinceX is strongly contractible, [ῡ] is invertible in KK H 0 C, C(X) for all compact subgroups H ⊆ G. That is,ῡ is a weak equivalence in the notation of [38]. It is shown in [10,38] that such maps induce isomorphisms on K top * (G, ).…”

Coarse and equivariant co-assembly maps

“…by a result of [2]. Now the assertion follows from the Five Lemma and the naturality of the K-theory long exact sequence for (9) as in [7].…”