Crossed modules and the integrability of Lie brackets

We show that the integrability obstruction of a transitive Lie algebroid coincides with the lifting obstruction of a crossed module of groupoids associated naturally with the given algebroid. Then we extend this result to general extensions of integrable transitive Lie algebroids by Lie algebra bundles. Such a lifting obstruction is directly related with the classification of extensions of transitive Lie groupoids. We also give a classification of such extensions which differentiates to the classification of transitive Lie algebroids discussed in \cite{KCHM:new}.Comment: 34 pages, revised version. New abstract and introduction. Added examples and remark

On a remark by Alan Weinstein

Alan Weinstein remarked that, working in the framework of diffeology, a construction from Noncommutative Differential Geometry might provide the non-trivial representations required for the geometric quantisation of a symplectic structure which is not integral. In this note we show that the construction we gave with P. Antonini does indeed provide non-trivial representations.Comment: 16 pages, to appear in Contemporary Mathematics, Proceedings of the AMS-EMS-SMF conference, special section on Diffeology, Grenoble 202

Integrable lifts for transitive Lie algebroids

Inspired by the work of Molino, we show that the integrability obstruction for transitive Lie algebroids can be made to vanish by adding extra dimensions. In particular, we prove that the Weinstein groupoid of a non-integrable transitive and abelian Lie algebroid is the quotient of a finite-dimensional Lie groupoid. Two constructions as such are given: First, explaining the counterexample to integrability given by Almeida and Molino, we see that it can be generalized to the construction of an \u201cAlmeida\u2013Molino\u201d integrable lift when the base manifold is simply connected. On the other hand, we notice that the classical de Rham isomorphism provides a universal integrable algebroid. Using it we construct a \u201cde Rham\u201d integrable lift for any given transitive Abelian Lie algebroid

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 ([H. E. Winkelnkemper, The graph of a foliation, Ann. Glob. Anal. Geom. 1 (3) (1983), 51-75.]); the same holds in the singular cases of [J. Pradines, How to define the differentiable graph of a singular foliation, C. Top. Geom. Diff. Cat. XXVI(4) (1985), 339-381.], [B. Bigonnet, J. Pradines, Graphe d'un feuilletage singulier, C. R. Acad. Sci. Paris 300 (13) (1985), 439-442.], [C. Debord, Local integration of Lie algebroids, Banach Center Publ. 54 (2001), 21-33.], [C. Debord, Holonomy groupoids of singular foliations, J. Diff. Geom. 58 (2001), 467-500.], 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 defining the foliation. Our groupoid is the quotient of germs of these bi-submersions with respect to an appropriate equivalence relatio