Survey on analytic and topological torsion

TL;DR

This survey reviews the concepts of analytic and topological torsion, their relationship via Cheeger-Müller theorem, and extensions to manifolds with boundary and group actions, highlighting how spectral data encodes topological invariants.

Contribution

It provides a comprehensive overview of analytic and topological torsion, including recent developments and generalizations beyond closed manifolds.

Findings

Cheeger-Müller theorem linking analytic and topological torsion

Extensions to manifolds with boundary

Analysis of torsion under finite group actions

Abstract

The article consists of a survey on analytic and topological torsion. Analytic torsion is defined in terms of the spectrum of the analytic Laplace operator on a Riemannian manifold, whereas topological torsion is defined in terms of a triangulation. The celebrated theorem of Cheeger and M\"uller identifies these two notions for closed Riemannian manifolds. We also deal with manifolds with boundary and with isometric actions of finite groups. The basic theme is to extract topological invariants from the spectrum of the analytic Laplace operator on a Riemannian manifold.