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

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)

Zbl 0575.58030
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