The Witten deformation for even dimensional spaces with cone-like singularities and admissible Morse functions

Ursula Ludwig

doi:10.1016/j.crma.2010.07.020

Differential Geometry/Differential Topology

The Witten deformation for even dimensional spaces with cone-like singularities and admissible Morse functions
[La déformation de Witten sur des espaces singuliers de dimension paire à singularités coniques]

Présenté par : Jean-Michel Bismut

Ursula Ludwig ¹

¹ Mathematisches Institut, Eckerstrasse 1, 79104 Freiburg, Germany

Comptes Rendus. Mathématique, Volume 348 (2010) no. 15-16, pp. 915-918.

Résumé
Abstract

Le but de cette Note est d'étendre la déformation de Witten au cas d'un espace singulier X de dimension paire à singularités coniques, muni de fonctions appelées fonctions de Morse admissibles. Comme conséquence on obtient des inégalités de Morse pour les nombres de Betti L² de X. La contribution d'un point singulier p de X aux inégalités de Morse s'exprime en fonction de la cohomologie d'intersection des données de Morse local. La définition des fonctions de Morse admissibles est inspirée par la théorie de Morse stratifiée de Goresky et MacPherson. Mais ici on travaille sur des espaces singuliers à singularités coniques au lieu d'espaces munis d'une stratification de Whitney.

In this Note we generalise the Witten deformation to even dimensional Riemannian manifolds with cone-like singularities X and certain functions f, which we call admissible Morse functions. As a corollary we get Morse inequalities for the L²-Betti numbers of X. The contribution of a singular point p of X to the Morse inequalities can be expressed in terms of the intersection cohomology of the local Morse datum of f at p. The definition of the class of functions which we study here is inspired by stratified Morse theory as developed by Goresky and MacPherson. However the setting here is different since the spaces considered here are manifolds with cone-like singularities instead of Whitney stratified spaces.

Reçu le : 2010-03-02
Accepté le : 2010-07-20
Publié le : 2010-08-01

DOI : 10.1016/j.crma.2010.07.020

Affiliations des auteurs :

Ursula Ludwig ¹

¹ Mathematisches Institut, Eckerstrasse 1, 79104 Freiburg, Germany

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Ursula Ludwig. The Witten deformation for even dimensional spaces with cone-like singularities and admissible Morse functions. Comptes Rendus. Mathématique, Volume 348 (2010) no. 15-16, pp. 915-918. doi : 10.1016/j.crma.2010.07.020. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.07.020/

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