Foliations, groupoids and Baum-Connes conjecture. (English. English summary) Zbl 1017.22002

Macho-Stadler, M.

Zap. Nauchn. Semin. POMI 266, Pt. II, 169-187 (2000).

This expository paper contains a statement of the Baum-Connes conjecture for foliated manifolds with detailed description of all necessary preliminaries: Lie groupoids, Morita-equivalence of groupoids, foliations defined by groupoids, homotopy and holonomy groupoids of a foliation, generalities on \(C^*\)-algebras, groupoid \(C^*\)-algebras, Morita equivalence of \(C^*\)-algebras, \(C^*\)-algebra of a foliation, \(K\)-theory of topological spaces and of \(C^*\)-algebras, \(K\)-orientation. Given a foliated manifold \((M,{\mathcal F})\) with the holonomy groupoid \(G\) the topological \(K\)-theory \(K_{\text{top}}(M/{\mathcal F})\) of the leaf space \(M/{\mathcal F}\) is defined as a generalized \(G\)-equivariant \(K\)-theory associated to the classifying space \(BG\) for \(G\) and the analytical \(K\)-theory \(K_{\text{an}}(M/{\mathcal F})\) of the leaf space is defined as \(K^*(C^*(M/{\mathcal F}))\) where \(C^*(M/{\mathcal F})\) denotes the \(C^*\)-algebra of the given foliated space \((M/{\mathcal F})\). The Baum-Connes conjecture asserts that the \(K\)-index map \[ \mu:K_{\text{top}}(M/{\mathcal F})\to K_{\text{an}}(M/{\mathcal F}) \] defined by the longitudinal index theorem [A. Connes and G. Skandalis, Publ. Res. Inst. Math. Sci. Kyoto Univ. 20, 1139-1183 (1984; Zbl 0575.58030)] is an isomorphism when the holonomy groups are torsion free. The author’s contribution to this field concerns the so-called reduced form of the Baum-Connes conjecture. The cases of foliations for which the Baum-Connes conjecture has already been verified are listed and commented.

Reviewer: E.S.Golod (Moskva)

MSC:

22A22

Topological groupoids (including differentiable and Lie groupoids)

Keywords:

foliated manifolds; Baum-Connes conjecture; Morita-equivalence of groupoids; Lie groupoids; groupoid \(C^*\)-algebras

Citations:

Zbl 0575.58030