[Paper Review] Survey on analytic and topological torsion

Wolfgang Lueck|arXiv (Cornell University)|Feb 26, 2015

Geometric and Algebraic Topology25 references3 citations

TL;DR

This survey provides a comprehensive, accessible overview of analytic and topological torsion, establishing their equivalence via the Cheeger-Müller theorem for closed Riemannian manifolds. It extends the theory to manifolds with boundary and finite group actions, introducing equivariant torsion invariants and connecting them to spectral invariants through zeta-regularized determinants and spectral density functions.

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 Müller 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.

Motivation & Objective

• To present a self-contained, accessible introduction to analytic and topological torsion for graduate students and non-experts.
• To explain how topological invariants can be extracted from the spectrum of the analytic Laplace operator on Riemannian manifolds.
• To unify the algebraic, topological, and analytic perspectives on torsion through the lens of spectral theory and zeta-regularization.
• To extend the theory to manifolds with boundary and to Riemannian manifolds with isometric finite group actions, introducing Poincaré torsion.
• To provide a foundation for understanding $L^2$-torsion and $L^2$-Betti numbers via von Neumann algebras and Fuglede-Kadison determinants.

Proposed method

• Reformulates classical invariants like determinant and trace using zeta-function regularization and spectral density functions for finite-dimensional Hilbert spaces.
• Applies the Hodge-de Rham decomposition to relate de Rham cohomology to harmonic forms on closed Riemannian manifolds.
• Defines topological torsion via cellular chain complexes and combinatorial Laplace operators, correcting Hilbert space structures via isomorphisms.
• Introduces analytic torsion via zeta-regularized determinants of the Laplace operator on $L^2$-forms, with spectral density functions as key tools.
• Establishes the Cheeger-Müller theorem by comparing analytic and topological torsion through spectral invariants and zeta-regularization.
• Extends the framework to equivariant settings using group actions, introducing Poincaré torsion as a new invariant in the presence of finite group actions.

Experimental results

Research questions

• RQ1 How can classical invariants like determinant and trace be generalized from finite-dimensional to infinite-dimensional Hilbert spaces using spectral theory?
• RQ2 What is the precise relationship between topological torsion (defined via triangulations) and analytic torsion (defined via the spectrum of the Laplace operator) on closed Riemannian manifolds?
• RQ3 How do boundary terms modify the Cheeger-Müller theorem for compact Riemannian manifolds with boundary?
• RQ4 What new torsion invariants arise in the presence of isometric finite group actions, and how do they relate to analytic and topological torsion?
• RQ5 How do $L^2$-invariants generalize analytic and topological torsion to non-compact coverings via von Neumann algebras and Fuglede-Kadison determinants?

Key findings

• The Cheeger-Müller theorem holds: analytic torsion and topological torsion are equal for closed Riemannian manifolds.
• For compact Riemannian manifolds with boundary, a correction term involving the Euler characteristic of the boundary is required to relate analytic and topological torsion.
• In the equivariant setting with a finite group action, three torsion invariants emerge: analytic, topological, and Poincaré torsion, with the latter not present in the non-equivariant case.
• For odd-dimensional, closed, oriented $G$-manifolds with free orientation-preserving $G$-action, the analytic torsion equals the topological torsion under the $ ho$-invariant map $ ho_{ ext{top}}$.
• For even-dimensional, closed, oriented $G$-manifolds, the analytic torsion vanishes, and the topological torsion is half the Poincaré torsion.
• The $L^2$-versions of torsion and Betti numbers are defined via spectral density functions and Fuglede-Kadison determinants in the context of group coverings and von Neumann algebras.