Foliations, groupoids and Baum-Connes conjecture
by
Marta Macho Stadler
Universidad del Pais Vasco-Euskal Herriko Unibertsitatea

In the ``non-commutative world", the most immediate and powerful tools are Homology and Fundamental Group. But, they have not ``non-commutative" obvious generalizations. Nevertheless, topological K-theory [A] is the most successful tool, since it pass easily to non-commutative world.

It is wellknown that, for each locally compact space M, the C*-algebra of continuous functions vanishing at infinity, C₀(M), allows us to ``reconstruct" M, and there is an isomorphism between the topological K-theory of M and the analytical K-theory of C₀(M).

The Baum-Connes conjecture, independently of its meaning in the context of index theory, looks for the establishment of an analogous of this isomorphism for some ``singular" spaces: the leaf spaces of foliated manifolds. Precisely, if F is a C^\infty-foliation on a manifold M, M/F is a bad quotient in many cases and thus, to obtain information about the transverse structure of the foliation, it is necessary to use another type of objects:

(i)

the dynamic of F is described by its holonomy groupoid G, which is a Lie groupoid. G can be considered as a desingularization of the leaf space M/F. As for any Lie groupoid, we can associate to G a C*-algebra of functions, C_red ^* (G), which is interpreted as the ``space of continuous functions vanishing at infinity" over M/F. And the analytical K-theory of the leaf space, K_an(M/F), is just defined as the K-theory of this C*-algebra, K _* (C_red ^* (G));

(ii)

moreover, we can construct a classifying space BG for G. G acts freely and properly on BG, which is not in general a manifold (and it has not even the homotopy type of a manifold!). BG can be considered as the leaf space, modulo homotopy. In [BC], the authors introduce a generalized G-equivariant K-theory associated to this object, K _{* , \tau} (BG), which is defined as the topological K-theory K_top(M/F), of the leaf space.

The proof of the conjecture would furnish us a relation between the information given by the transverse structure of the foliation (through G) and the geometry granted by BG; in other words, it would ``present" a geometrical interpretation of the analytical object K _* (C_red ^* (G)).

In this talk, we describe the principal notions involved in the statement of the conjecture, and we point out the actual status of it.

References
[A] M.F. Atiyah, K-theory, Advanced Book Classics Series, Addison-Wesley Pub. Co., Inc, 1989.

[BC] P. Baum and A. Connes, Geometric K-theory for Lie groups and foliations, preprint, 1982.

[C] A. Connes, Géométrie non commutative, InterEditions, Paris, 1990.

[HM] G. Hector et M. Macho Stadler, Isomorphisme de Thom pour les feuilletages presque sans holonomie, Comptes Rendues de l'Académie des Sciences, Série I, 325 (9), 1015-1018, 1998.

[M1] M. Macho Stadler, La Conjecture de Baum-Connes pour un feuilletage sans holonomie de codimension un sur une variété fermée, Publicacions Matemàtiques 33, 445-457, 1989.

[M2] M. Macho Stadler, Isomorphisme de Thom et feuilletages presque sans holonomie, C. R. Acad. Sci. Paris 325, Série I, 1015-1018, 1997.

[MO] M. Macho Stadler and M. O'uchi, Correspondances of groupoid C*-algebras, to appear in Journal of Operator Theory, 1999.

Date received: June 2, 1999