Foliations, groupoids and Baum-Connes conjecture

Baum–Connes conjecture looks for the establishment of an analogue of the well-known isomorphism between the topological K-theory of a locally compact space $M$ and the analytical K-theory of the C*-algebra of continuous functions on $M$ vanishing at infinity, for the leaf spaces of foliated manifolds. In this work we describe the principal notions involved in the statement of the conjecture, and we point out the actual status of it.