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An index formula for the intersection Euler characteristic of an infinite cone

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Abstract

The aim of this article is to establish an index formula for the intersection Euler characteristic of a cone. The main actor 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.

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Acknowledgements

First and foremost, I would like to thank Jean-Michel Bismut for many discussions and for his continuous and patient support. I have profited from discussions with Juan Gil, Thomas Krainer and Gerardo Mendoza during a one week stay in Altoona and Philadelphia in summer 2014. Many thanks to the participants in the “Groupe de travail sur l’opérateur de Dirac” Paolo Antonini, Sara Azzali, Bo Liu, Xiaonan Ma, Martin Puchol, Nikhil Savale and Shu Shen. The author has been supported by the Marie Curie Intra European Fellowship (within the 7th European Community Framework Programme) COMPTORSING and wishes to thank the Département de Mathématiques, Université Paris-Orsay, for hospitality.

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  1. Universität Duisburg-Essen, Fakultät für Mathematik, 45117, Essen, Germany

    Ursula Ludwig

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Ludwig, U. An index formula for the intersection Euler characteristic of an infinite cone. Math. Z. 296, 99–126 (2020). https://doi.org/10.1007/s00209-019-02423-5

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