The holonomy groupoid of a singular foliation

We construct the holonomy groupoid of any singular foliation. In the regular case this groupoid coincides with the usual holonomy groupoid of Winkelnkemper ([30]); the same holds in the singular cases of [24], [2], [9], [10], which from our point of view can be thought of as being "almost regular''. In the general case, the holonomy groupoid can be quite an ill behaved geometric object. On the other hand it often has a nice longitudinal smooth structure. Nonetheless, we use this groupoid to generalize to the singular case Connes' construction of the C*-algebra of the foliation. We also outline the construction of a longitudinal pseudo-differential calculus; the analytic index of a longitudinally elliptic operator takes place in the K-theory of our C*-algebra. In our construction, the key notion is that of a bi-submersion which plays the role of a local Lie groupoid de. ning the foliation. Our groupoid is the quotient of germs of these bi-submersions with respect to an appropriate equivalence relation.