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A geometric description of equivariant K-homology for proper actions. (English) Zbl 1216.19006
Blanchard, Etienne (ed.) et al., Quanta of maths. Conference on non commutative geometry in honor of Alain Connes, Paris, France, March 29–April 6, 2007. Providence, RI: American Mathematical Society (AMS); Cambridge, MA: Clay Mathematics Institute (ISBN 978-0-8218-5203-3/pbk). Clay Mathematics Proceedings 11, 1-22 (2010).
Summary: Let \(\mathcal G\) be a discrete group and let \(X\) be a \(\mathcal G\)-finite, proper \(\mathcal G\)-\(\mathcal{CW}\)-complex. We prove that Kasparov’s equivariant \(\mathcal K\)-homology groups \(\mathcal{KK}^G({\mathcal C}_0(X),\mathbb C)\) are isomorphic to the geometric equivariant \(\mathcal K\)-homology groups of \(X\) that are obtained by making the geometric \(\mathcal K\)-homology theory of Baum and Douglas equivariant in the natural way. This reconciles the original and current formulations of the Baum-Connes conjecture for discrete groups.
For the entire collection see [Zbl 1206.00042].

19K33 Ext and \(K\)-homology
19K35 Kasparov theory (\(KK\)-theory)
19L47 Equivariant \(K\)-theory
58J22 Exotic index theories on manifolds