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 Γ , the direct limit of the uniform K-homology of the Rips complexes of Γ , limd→∞Ku ∗ (PdΓ ), is isomorphic to K top ∗ (Γ, ∞Γ ), the left-hand side of the Baum–Connes conjecture with coefficients in ∞Γ . 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. © 2009 Elsevier Inc. All rights reserved.