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Comptes Rendus Mathématique
Volume 348, n° 15-16
pages 915-918 (août 2010)
Doi : 10.1016/j.crma.2010.07.020
Received : 2 Mars 2010 ;  accepted : 20 July 2010
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
 

Ursula Ludwig
Mathematisches Institut, Eckerstrasse 1, 79104 Freiburg, Germany 

Abstract

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.

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Résumé

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.

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