Synopsis "introduction to the baum-connes conjecture"
the baum-connes conjecture is part of a. connes non-commutative geometry programme. it can be viewed as a conjectural generalisation of the atiyah-singer index theorem, to the equivariant setting (the ambient manifold is not compact, but some compactness is restored by means of a proper, co-compact action of a group gamma). like the atiyah-singer theorem, the baum-connes conjecture states that a purely topological object coincides with a purely analytical one. for a given group gamma, the topological object is the equivariant k-homology of the classifying space for proper actions of gamma, while the analytical object is the k-theory of the c*-algebra associated with gamma in its regular representation. the baum-connes conjecture implies several other classical conjectures, ranging from differential topology to pure algebra. it has also strong connections with geometric group theory, as the proof of the conjecture for a given group gamma usually depends heavily on geometric properties of gamma. this book is intended for graduate students and researchers in geometry (commutative or not), group theory, algebraic topology, harmonic analysis, and operator algebras. it presents, for the first time in book form, an introduction to the baum-connes conjecture. it starts by defining carefully the objects in both sides of the conjecture, then the assembly map which connects them. thereafter it illustrates the main tool to attack the conjecture (kasparovs theory), and it concludes with a rough sketch of v. lafforgues proof of the conjecture for co-compact lattices in in spn1, sl(3r), and sl(3c).
