Uniform K-homology theory
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 ?, the direct limit of the uniform K-homology of the Rips complexes of ?, is isomorphic to the left-hand side of the Baum–Connes conjecture with coefficients in ???. 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.
analytic k-homology, coarse assembly map, uniform roe algebra
88-121
Spakula, Jan
c43164e4-36a7-4372-9ce2-9bfbba775d77
1 July 2009
Spakula, Jan
c43164e4-36a7-4372-9ce2-9bfbba775d77
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 ?, the direct limit of the uniform K-homology of the Rips complexes of ?, is isomorphic to the left-hand side of the Baum–Connes conjecture with coefficients in ???. 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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Published date: 1 July 2009
Keywords:
analytic k-homology, coarse assembly map, uniform roe algebra
Organisations:
Pure Mathematics
Identifiers
Local EPrints ID: 358938
URI: http://eprints.soton.ac.uk/id/eprint/358938
ISSN: 0022-1236
PURE UUID: 1aca9a93-db02-4cae-8e76-f3537968a801
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Date deposited: 15 Oct 2013 15:25
Last modified: 15 Mar 2024 03:48
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