The Baum-Connes conjecture: an extended survey (original) (raw)

2019, arXiv (Cornell University)

To Alain Connes, for providing lifelong inspiration * (G), is called the geometric, or topological side. This is actually misleading, as its definition is awfully analytic, involving Kasparov's bivariant theory (see Chapter 3). A better terminology would be the commutative side, as indeed it involves a space EG, the classifiying space for proper actions of G (see Chapter 4), and K top * (G) is the G-equivariant K-homology of EG. When G is discrete and torsion-free, then EG = EG = BG, the universal cover of the classifying space BG. As G acts freely on EG, the G-equivariant K-homology of EG is K * (BG), the ordinary K-homology of BG, where Khomology for spaces can be defined as the homology theory dual to topological K-theory for spaces. • The assembly map µ r will be defined in Chapter 4 using Kasparov's equivariant KK-theory. Let us only give here a flavor of the meaning of this map. It was discovered in the late 1970's and early 1980's that the K-theory group K * (C * r (G)) is a receptacle for indices, see section 2.3. More precisely, if M is a smooth manifold with a proper action of G and compact quotient, and D an elliptic G-invariant differential operator on M , then D has an index ind G (D) living in K * (C * r (G)). Therefore, the geometric group K top * (G) should be thought of as the set of homotopy classes of such pairs (M, D), and the assembly map µ r maps the class [(M, D)] to ind G (D) ∈ K * (C * r (G)).