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The holonomy groupoid of a singular foliation


Androulidakis, I; Skandalis, G (2009). The holonomy groupoid of a singular foliation. Journal für die Reine und Angewandte Mathematik, 626:1-37.

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), 51–75.]); 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.

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.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:January 2009
Deposited On:11 Nov 2009 13:52
Last Modified:05 Apr 2016 13:31
Publisher:De Gruyter
ISSN:0075-4102
Publisher DOI:https://doi.org/10.1515/CRELLE.2009.001
Related URLs:http://arxiv.org/abs/math/0612370
http://www.ams.org/mathscinet-getitem?mr=2492988

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