A short course on Witten Helffer-Sj\"ostrand theory

TL;DR

This paper introduces Witten-Helffer-Sj"ostrand theory, an advanced mathematical framework that enhances Morse and Hodge-de Rham theories by integrating spectral theory, with applications in topology, geometry, and dynamics.

Contribution

It provides a comprehensive overview of Witten-Helffer-Sj"ostrand theory, highlighting its improvements over classical theories and its versatile applications in various mathematical fields.

Findings

Enhanced comparison of numerical invariants using spectral theory.

Refined approach to Novikov Morse theory.

Applications in symplectic topology and dynamics.

Abstract

Witten-Helffer-Sj\"ostrand theory is an addition to Morse theory and Hodge-de Rham theory for Riemannian manifolds and considerably improves on them by injecting some spectral theory of elliptic operators. It can serve as a general tool to prove results about comparison of numerical invariants associated to compact manifolds analytically, i.e. by using a Riemannian metric, or combinatorially, i.e. by using a triangulation. It can be also refined to provide an alternative presentation of Novikov Morse theory and improve on it in many respects. In particular it can be used in symplectic topology and in dynamics. This material represents my Notes for a three lectures course given at the Goettingen summer school on groups and geometry, June 2000.