Formulation of the Baum-Connes Conjecture

The Baum-Connes conjecture is most easily formulated for those discrete, torsion-free groups G. Suppose that M is a closed, spin^c manifold and that W is a principal G-space over M. The Dirac operator on M lifts to a G-equivariant Dirac operator on W. Because W is noncompact this lifted operator does not have an integer Fredholm index, but it does have a "higher index" which lies in the K-theory of the reduced C*-algebra of G. Now as Atiyah showed and Baum made precise, spin^c-manifolds equipped with principal G-covers may be viewed as cycles for the K-homology of the classifying space BG. The higher index construction gives rise to an "assembly map" from the K-homology of BG to the K-theory of the reduced C*-algebra of G. The Baum-Connes conjecture asserts that this map is an isomorphism. Roughly speaking, the conjecture asserts that every element in the K-theory group of a reduced group C*-algebra is a higher index of a Dirac operator, and that the only relations among these indices are geometric in nature (for example, bordism of manifolds gives rise to an identity of higher indices, just as it gives rise to an identity between oridinary Fredholm indices).

If G is not torsion-free (or not even discrete), then the formulation is a bit more complex. This is because there are new ways of obtaining higher indices in the K-theory of a reduced group C*-algebra: every equivariant operator on a proper (not necessarily principal), G-compact manifold has an index in the K-theory of the group C*-algebra. The collection of all such index problems may be organized into the equivariant K-homology of the classifying space for proper G-actions, and the Baum-Connes conjecture asserts that the natural assembly map from this equivariant K-homology group into the K-theory of the group C*-algebra is an isomorphism.

In the special case of connected Lie groups, the Baum-Connes conjecture can be made into a more precise statement about Dirac induction. This is the Connes-Kasparov conjecture.

Selected References

Balmer, Paul; Matthey, Michel. Model theoretic reformulation of the Baum-Connes and Farrell-Jones conjectures. Adv. Math. 189 (2004), no. 2, 495-500.   MR
Baum, Paul; Connes, Alain. Geometric K-theory for Lie groups and foliations. Enseign. Math. (2) 46 (2000), no. 1-2, 3-42.   MR
Baum, Paul; Connes, Alain; Higson, Nigel. Classifying space for proper actions and K-theory of group C*-algebras. C*-algebras: 1943-1993 (San Antonio, TX, 1993), 240-291, Contemp. Math., 167, Amer. Math. Soc., Providence, RI, 1994.   MR
Davis, James F.; Lueck, Wolfgang. Spaces over a category and assembly maps in isomorphism conjectures in K- and L-theory. K-Theory 15 (1998), no. 3, 201-252.   MR
Meyer, Ralf; Nest, Ryszard. The Baum-Connes conjecture via localisation of categories. Topology 45 (2006), no. 2, 209-259.   MR
Mislin, Guido; Valette, Alain. Proper group actions and the Baum-Connes conjecture. Advanced Courses in Mathematics. CRM Barcelona. Birkhaeuser Verlag, Basel, 2003. viii+131 pp. ISBN: 3-7643-0408-1   MR
Roe, John. Comparing analytic assembly maps. Q. J. Math. 53 (2002), no. 2, 241-248.   MR
Valette, Alain. Introduction to the Baum-Connes conjecture. From notes taken by Indira Chatterji. With an appendix by Guido Mislin. Lectures in Mathematics ETH Zuerich. Birkhaeuser Verlag, Basel, 2002. x+104 pp. ISBN: 3-7643-6706-7   MR
Valette, Alain. On the Baum-Connes assembly map for discrete groups. With an appendix by Dan Kucerovsky. Adv. Courses Math. CRM Barcelona, Proper group actions and the Baum-Connes conjecture, 79-124, Birkhaeuser, Basel, 2003.   MR

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