The Analytic Torsion of the cone over an odd dimensional manifold

TL;DR

This paper investigates the analytic torsion of cones over odd-dimensional manifolds, revealing its decomposition into boundary and topological terms, and establishing connections with the Cheeger-Muller theorem and Poincaré duality.

Contribution

It provides a decomposition formula for the analytic torsion of cones over odd-dimensional manifolds and links it to boundary invariants and the Cheeger-Muller theorem.

Findings

Decomposition of analytic torsion into boundary and topological parts

Validation of Cheeger-Muller theorem for cones over odd spheres

Establishment of Poincaré duality for cone torsion

Abstract

We study the analytic torsion of the cone over an orientable odd dimensional compact connected Riemannian manifold W. We prove that the logarithm of the analytic torsion of the cone decomposes as the sum of the logarithm of the root of the analytic torsion of the boundary of the cone, plus a topological term, plus a further term that is a rational linear combination of local Riemannian invariants of the boundary. We also prove that this last term coincides with the anomaly boundary term appearing in the Cheeger Muller theorem for a manifold with boundary, according to Bruning and Ma, either in the case that W is an odd sphere or has dimension smaller than six. It follows in particular that the Cheeger Muller theorem holds for the cone over an odd dimensional sphere. We also prove Poincare duality for the analytic torsion of a cone.