The holonomy groupoid of a singular foliation

By Iakovos Androulidakis, G. Skandalis. 2006

144 citations recorded by the source corpus.

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), 51–75.]); 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 relation.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a8e57662823210a0c1a399a86fdda2de13bad056

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In Algebra and Number Theory

1. Iakovos Androulidakis
2. G. Skandalis
3. J. Lescure
4. Frédéric Rochon
5. C. Debord
6. J. Tu

In Mathematical Physics

1. Iakovos Androulidakis
2. G. Skandalis
3. J. Lescure
4. Frédéric Rochon
5. C. Debord
6. J. Tu

Citations in the corpus

• Pseudodifferential operators on manifolds with fibred corners (2011, background): “Moreover it defines a singular folaliation in the sense of [4] which is almost regular as the foliations studied by the first author in [11].”
• Pseudodifferential calculus on a singular foliation (2009, background): “A key notion in [1] is that of a bi-submersion which will be also of importance here.”