Classifying Spaces and the Baum-Connes Conjecture

The Baum-Connes conjecture identifies the K-theory of the reduced C*-algebra of a group G with the equivariant K-theory of the classifying space for proper G-actions through a so-called assembly map from K-homology to K-theory. This classifying space is a proper G-space with the feature that any other proper G-space may be equivariantly mapped into it in a unique-up-to-equivariant-homotopy way. In many situations the classifying space for proper actions may be concretely identified. For example for a semisimple Lie group it is the symmetric space G/K and for a semisimple p-adic group it is the Bruhat-Tits affine building.

For discrete groups the classifying space for proper actions has been studied carefully from the point of view of equivariant homotopy theory, and this has led to knew views on the Baum-Connes assembly map in this case.

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
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
Lueck, Wolfgang. Survey on classifying spaces for families of subgroups. Infinite groups: geometric, combinatorial and dynamical aspects, 269-322, Progr. Math., 248, Birkhaeuser, Basel, 2005.   MR
Lueck, Wolfgang. The relation between the Baum-Connes conjecture and the trace conjecture. Invent. Math. 149 (2002), no. 1, 123-152.   MR
Lueck, Wolfgang. The type of the classifying space for a family of subgroups. J. Pure Appl. Algebra 149 (2000), no. 2, 177-203.   MR
Lueck, Wolfgang; Reich, Holger. The Baum-Connes and the Farrell-Jones conjectures in K- and L-theory. Handbook of K-theory. Vol. 1, 2, 703-842, Springer, Berlin, 2005.   MR
Mislin, Guido. Equivariant K-homology of the classifying space for proper actions. Proper group actions and the Baum-Connes conjecture, 1-78, Adv. Courses Math. CRM Barcelona, Birkhaeuser, Basel, 2003.   MR

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