Differential geometry
An index formula for the intersection Euler characteristic of an infinite cone
Comptes Rendus. Mathématique, Volume 355 (2017) no. 1, pp. 94-98.

The aim of this note is to establish an index formula for the intersection Euler characteristic of a cone. The main actor of these notes is the model Witten Laplacian on the infinite cone. First, we study its spectral properties and establish a McKean–Singer-type formula. We also give an explicit formula for the zeta function of the model Witten Laplacian. In a second step, we apply local index techniques to the model Witten Laplacian. By combining these two steps, we express the absolute and relative intersection Euler characteristic of the cone as a sum of two terms, a term which is local, and a second term which is the Cheeger invariant.

Le but de cette note est d'établir un théorème de l'indice pour la caractéristique d'Euler d'intersection d'un cône. Dans un premier temps, on étudie les propriétés spectrales du laplacien de Witten et on établit une formule de McKean–Singer. On donne aussi une formule explicite pour la fonction zêta associée au laplacien de Witten. Dans une deuxième partie, on applique des techniques d'indice local au laplacien de Witten. On obtient une formule qui exprime la caractéristique d'Euler d'intersection du cône comme la somme de deux termes, l'un local, l'autre étant l'invariant de Cheeger.

Accepted:
Published online:
DOI: 10.1016/j.crma.2016.11.003

Ursula Ludwig 1

1 Universität Duisburg–Essen, Fakultät für Mathematik, 45117 Essen, Germany
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Ursula Ludwig. An index formula for the intersection Euler characteristic of an infinite cone. Comptes Rendus. Mathématique, Volume 355 (2017) no. 1, pp. 94-98. doi : 10.1016/j.crma.2016.11.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.11.003/

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