Uniform K-homology theory |
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Authors: | Ján Špakula |
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Institution: | Mathematisches Institut, Universität Münster, Einsteinstr. 62, 48149 Münster, Germany |
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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, , is isomorphic to , 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. |
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Keywords: | primary 46L80 |
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