Cheeger–Müller and Bismut–Zhang theorem on singular spaces

Abstract

In this article, we present the Cheeger–Müller/Bismut–Zhang theorem for singular spaces with isolated conical singularities proved recently by the author. The classical Cheeger–Müller theorem, the comparison of analytic and topological torsion of a smooth compact manifold, is one of the most important comparison theorems in global analysis. Its most general version for smooth compact manifolds has been achieved by Bismut and Zhang by combining local index techniques and the Witten deformation.

The question of a Cheeger–Müller theorem for spaces with isolated conical singularities has been raised by Dar in the late 1980s, very early in the development of the (global) analysis on these spaces. The Cheeger–Müller/Bismut–Zhang theorem for singular spaces with isolated conical singularities presented here answers the question by generalising Bismut and Zhang’s approach to the singular setting. An important step in this programme is hence the generalisation of the Morse–Thom–Smale complex and the Witten deformation to the singular setting, achieved by the author, and which we also review here.

This article is a written and extended version of the author’s invited talk at the 9ECM, Sevilla, 2024.