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.
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@article{CRMATH_2013__351_15-16_613_0,
author = {Debord, Claire},
title = {Longitudinal smoothness of the holonomy groupoid},
journal = {Comptes Rendus. Math\'ematique},
pages = {613--616},
publisher = {Elsevier},
volume = {351},
number = {15-16},
year = {2013},
doi = {10.1016/j.crma.2013.07.025},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/j.crma.2013.07.025/}
}
TY - JOUR AU - Debord, Claire TI - Longitudinal smoothness of the holonomy groupoid JO - Comptes Rendus. Mathématique PY - 2013 SP - 613 EP - 616 VL - 351 IS - 15-16 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2013.07.025/ DO - 10.1016/j.crma.2013.07.025 LA - en ID - CRMATH_2013__351_15-16_613_0 ER -
%0 Journal Article %A Debord, Claire %T Longitudinal smoothness of the holonomy groupoid %J Comptes Rendus. Mathématique %D 2013 %P 613-616 %V 351 %N 15-16 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2013.07.025/ %R 10.1016/j.crma.2013.07.025 %G en %F CRMATH_2013__351_15-16_613_0
Debord, Claire. 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. http://archive.numdam.org/articles/10.1016/j.crma.2013.07.025/
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