Lectures on Witten Helffer Sj\"ostrand Theory

TL;DR

Witten-Helffer-Sj"ostrand theory extends De Rham-Hodge theory to compare numerical invariants of manifolds using analytic and combinatorial methods, with applications in topology and $L_2$-torsion.

Contribution

This paper introduces a geometric interpretation of WHS theory via Morse functions and stable manifolds, providing new proofs and results in topology.

Findings

New proofs for difficult topological results

Positive solution for an $L_2$-torsion conjecture

Analytic approach to manifold invariants

Abstract

Witten- Helffer-Sj\"ostrand theory is a considerable addition to the De Rham- Hodge theory for Riemannian manifolds and 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. In this presentation a triangulation, or a partition of a smooth manifold in cells, will be viewed in a more analytic spirit, being provided by the stable manifolds of the gradient of a nice Morse function. WHS theory was recently used both for providing new proofs for known but difficult results in topology, as well as new results and a positive solution for an important conjecture about $L_2-$torsion, cf [BFKM]. This presentation is a short version of a one quarter course I have given during the spring of 1997 at OSU.