Longitudinal smoothness of the holonomy groupoid

Claire Debord

doi:10.1016/j.crma.2013.07.025

Functional Analysis/Differential Geometry

Longitudinal smoothness of the holonomy groupoid

Presented by: the Editorial Board

Claire Debord ¹

¹ Université Blaise-Pascal, laboratoire de mathématiques UMR 6620 CNRS, campus des Cézeaux, BP 80026, 63171 Aubière cedex, France

Comptes Rendus. Mathématique, Volume 351 (2013) no. 15-16, pp. 613-616.

Abstract (Original language)
Résumé

Iakovos Androulidakis and Georges Skandalis have defined a holonomy groupoid for any singular foliation. This groupoid, whose topology is usually quite bad, is the starting point for the study of longitudinal pseudodifferential calculus on such foliation and its associated index theory. These studies can be highly simplified under the assumption of the holonomy groupoid being longitudinally smooth. In this note, we rephrase the period bounding lemma that asserts that a vector field on a compact manifold admits a strictly positive lower bound for its periodic orbits in order to prove that the holonomy groupoid is always longitudinally smooth.

Iakovos Androulidakis et Georges Skandalis ont défini un groupoïde dʼholonomie pour tout feuilletage singulier. Ce groupoïde, dont la topologie est généralement assez singulière, est le point de départ dʼun calcul pseudodifferentiel longitudinal ainsi que dʼune théorie de lʼindice pour de tels feuilletages. Ces travaux sont grandement simplifiés sous lʼhypothèse de différentiabilité longitudinale du groupoïde dʼholonomie. Dans cette note, nous réinterprétons le period bounding lemma pour montrer que le groupoïde dʼholonomie est toujours longitudinalement lisse.

Received: 2013-07-19
Accepted: 2013-07-29
Published online: 2013-08-01

DOI: 10.1016/j.crma.2013.07.025

Author's affiliations:

Claire Debord ¹

¹ Université Blaise-Pascal, laboratoire de mathématiques UMR 6620 CNRS, campus des Cézeaux, BP 80026, 63171 Aubière cedex, France

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Claire Debord. Longitudinal smoothness of the holonomy groupoid. Comptes Rendus. Mathématique, Volume 351 (2013) no. 15-16, pp. 613-616. doi : 10.1016/j.crma.2013.07.025. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.07.025/

References
Cited by

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[2] I. Androulidakis; G. Skandalis The holonomy groupoid of a singular foliation, J. Reine Angew. Math., Volume 626 (2009), pp. 1-37

[3] I. Androulidakis; M. Zambon Smoothness of holonomy covers for singular foliations and essential isotropy, Math. Z. (2013) (in press) | arXiv

[4] M. Crainic; R.L. Fernandes Integrability of Lie brackets, Ann. of Math. (2), Volume 157 (2003) no. 2, pp. 575-620

[5] K. Mackenzie Lie Groupoids and Lie Algebroids in Differential Geometry, London Mathematical Society Lecture Note Series, vol. 124, Cambridge University Press, Cambridge, 1987

[6] V. Ozols Critical points of the length of a Killing vector field, J. Differ. Geom., Volume 7 (1972), pp. 143-148

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