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
[Différentiabilité longitudinale du groupoïde dʼholonomie]

Présenté par : 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 (VO)
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.

Reçu le : 2013-07-19
Accepté le : 2013-07-29
Publié le : 2013-08-01

DOI : 10.1016/j.crma.2013.07.025

Affiliations des auteurs :

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/

Bibliographie
Cité par

[1] R. Abraham; J. Robbin Transversal Mappings and Flows, W.A. Benjamin, Inc., New York, Amsterdam, 1967

[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

Wayne M. Lawton; August K. Tsikh Fourier Quasicrystals on $R^{n}$ , The Journal of Geometric Analysis, Volume 35 (2025) no. 3 | DOI:10.1007/s12220-025-01911-x
Omar Mohsen Tangent groupoid and tangent cones in sub-Riemannian geometry, Duke Mathematical Journal, Volume 173 (2024) no. 16, pp. 3179-3218 | DOI:10.1215/00127094-2024-0013 | Zbl:7974310
Alfonso Garmendia; Joel Villatoro Integration of Singular Foliations via Paths, International Mathematics Research Notices, Volume 2022 (2022) no. 23, p. 18401 | DOI:10.1093/imrn/rnab177
Camille Laurent-Gengoux; Leonid Ryvkin The holonomy of a singular leaf, Selecta Mathematica. New Series, Volume 28 (2022) no. 2, p. 38 (Id/No 45) | DOI:10.1007/s00029-021-00753-z | Zbl:1491.53031
Camille Laurent-Gengoux; Sylvain Lavau; Thomas Strobl The universal Lie $\infty$ -algebroid of a singular foliation, Documenta Mathematica, Volume 25 (2020), pp. 1571-1652 | DOI:10.25537/dm.2020v25.1571-1652 | Zbl:1453.53033
Alfonso Garmendia; Marco Zambon Hausdorff Morita equivalence of singular foliations, Annals of Global Analysis and Geometry, Volume 55 (2019) no. 1, pp. 99-132 | DOI:10.1007/s10455-018-9620-6 | Zbl:1415.53014
Iakovos Androulidakis; Georges Skandalis A Baum-Connes conjecture for singular foliations, Annals of $K$ -Theory, Volume 4 (2019) no. 4, pp. 561-620 | DOI:10.2140/akt.2019.4.561 | Zbl:1437.19002
Iakovos Androulidakis; Marco Zambon Almost regular Poisson manifolds and their holonomy groupoids, Selecta Mathematica. New Series, Volume 23 (2017) no. 3, pp. 2291-2330 | DOI:10.1007/s00029-017-0319-5 | Zbl:1380.53032
Claire Debord; Georges Skandalis Stability of Lie groupoid $C^{*}$ -algebras, Journal of Geometry and Physics, Volume 105 (2016), pp. 66-74 | DOI:10.1016/j.geomphys.2016.03.011 | Zbl:1339.58007
Yves F. Meyer Measures with locally finite support and spectrum, Proceedings of the National Academy of Sciences of the United States of America, Volume 113 (2016) no. 12, pp. 3152-3158 | DOI:10.1073/pnas.1600685113 | Zbl:1367.28002
Nir Lev; Alexander Olevskii Quasicrystals and Poisson’s summation formula, Inventiones mathematicae, Volume 200 (2015) no. 2, p. 585 | DOI:10.1007/s00222-014-0542-z
Iakovos Androulidakis; Marco Zambon Holonomy transformations for singular foliations, Advances in Mathematics, Volume 256 (2014), pp. 348-397 | DOI:10.1016/j.aim.2014.02.003 | Zbl:1294.53025

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