[Inégalités de Morse–Bott, orientations et isomorphisme de Thom en homologie de Morse]
Les inégalités de Morse–Bott relient la topologie d'une variété compacte sans bord à celle de l'ensemble des points critiques d'une fonction de Morse–Bott définie sur cette variété. Elles sont parfois énoncées sous des hypothèses d'orientation inexactes. Nous montrons que ces hypothèses sont insuffisantes grâce à un contre-exemple explicite et clarifions l'origine de cette erreur.
The Morse–Bott inequalities relate the topology of a closed manifold to the topology of the critical point set of a Morse–Bott function defined on it. The Morse–Bott inequalities are sometimes stated under incorrect orientation assumptions. We show that these assumptions are insufficient with an explicit counterexample and clarify the origin of the mistake.
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@article{CRMATH_2016__354_10_1026_0,
author = {Rot, Thomas O.},
title = {The {Morse{\textendash}Bott} inequalities, orientations, and the {Thom} isomorphism in {Morse} homology},
journal = {Comptes Rendus. Math\'ematique},
pages = {1026--1028},
publisher = {Elsevier},
volume = {354},
number = {10},
year = {2016},
doi = {10.1016/j.crma.2016.08.003},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2016.08.003/}
}
TY - JOUR AU - Rot, Thomas O. TI - The Morse–Bott inequalities, orientations, and the Thom isomorphism in Morse homology JO - Comptes Rendus. Mathématique PY - 2016 SP - 1026 EP - 1028 VL - 354 IS - 10 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2016.08.003/ DO - 10.1016/j.crma.2016.08.003 LA - en ID - CRMATH_2016__354_10_1026_0 ER -
%0 Journal Article %A Rot, Thomas O. %T The Morse–Bott inequalities, orientations, and the Thom isomorphism in Morse homology %J Comptes Rendus. Mathématique %D 2016 %P 1026-1028 %V 354 %N 10 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2016.08.003/ %R 10.1016/j.crma.2016.08.003 %G en %F CRMATH_2016__354_10_1026_0
Rot, Thomas O. The Morse–Bott inequalities, orientations, and the Thom isomorphism in Morse homology. Comptes Rendus. Mathématique, Tome 354 (2016) no. 10, pp. 1026-1028. doi : 10.1016/j.crma.2016.08.003. http://www.numdam.org/articles/10.1016/j.crma.2016.08.003/
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