Uniform K-homology theory

We define a uniform version of analytic K-homology theory for separable, proper metric spaces. Furthermore, we define an index map from this theory into the K-theory of uniform Roe C*-algebras, analogous to the coarse assembly map from analytic K-homology into the K-theory of Roe C*-algebras. We show that our theory has a Mayer-Vietoris sequence. We prove that for a torsion-free countable discrete group Gamma, the direct limit of the uniform K-homology of the Rips complexes of Gamma, lim(d ->infinity) K(*)(u)(P(d) Gamma), is isomorphic to K(*)(top)(Gamma, l(infinity) Gamma), the left-hand side of the Baum-Connes conjecture with coefficients in l(infinity)Gamma. In particular, this provides a computation of the uniform K-homology groups for some torsion-free groups. As an application of uniform K-homology, we prove a criterion for amenability in terms of vanishing of a "fundamental class", in spirit of similar criteria in uniformly finite homology and K-theory of uniform Roe algebras. (c) 2009 Elsevier Inc. All rights reserved.