@article {11483, title = {The holonomy groupoid of a singular foliation}, journal = {J. Reine Angew. Math.}, volume = {626}, year = {2009}, pages = {1-37}, abstract = {

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{\textendash}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{\textendash}381.], [B. Bigonnet, J. Pradines, Graphe d{\textquoteright}un feuilletage singulier, C. R. Acad. Sci. Paris\ 300\ (13) (1985), 439{\textendash}442.], [C. Debord, Local integration of Lie algebroids, Banach Center Publ.\ 54\ (2001), 21{\textendash}33.], [C. Debord, Holonomy groupoids of singular foliations, J. Diff. Geom.\ 58\ (2001), 467{\textendash}500.], which from our point of view can be thought of as being {\textquotedblleft}almost regular{\textquotedblright}. 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{\textquoteright} 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 relation.

}, url = {https://www.degruyter.com/view/j/crll.2009.2009.issue-626/crelle.2009.001/crelle.2009.001.xml}, author = {Iakovos Androulidakis and Georges Skandalis} }