Uniform K-homology theory

Abstract

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, lim d → ∞ K * u ( P d Γ ) , is isomorphic to K * top ( Γ , ℓ ∞ Γ ) , the left-hand side of the Baum–Connes conjecture with coefficients in ℓ ∞ 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.