A Cheeger-M\"uller theorem for manifolds with wedge singularities

TL;DR

This paper extends the Cheeger-Müller theorem to manifolds with wedge singularities by analyzing the spectrum and heat kernel of the Hodge Laplacian, establishing a relation between analytic and Reidemeister torsion in this setting.

Contribution

It provides a uniform construction of the resolvent and heat kernel on manifolds with corners under spectral gap conditions, and proves a Cheeger-Müller theorem for wedge singularities.

Findings

Uniform resolvent and heat kernel constructions on manifolds with corners.

A Cheeger-Müller theorem relating analytic and Reidemeister torsion for wedge singularities.

Conditions under which the theorem holds in odd dimensions.

Abstract

We study the spectrum and heat kernel of the Hodge Laplacian with coefficients in a flat bundle on a closed manifold degenerating to a manifold with wedge singularities. Provided the Hodge Laplacians in the fibers of the wedge have an appropriate spectral gap, we give uniform constructions of the resolvent and heat kernel on suitable manifolds with corners. When the wedge manifold and the base of the wedge are odd dimensional, this is used to obtain a Cheeger-M\"uller theorem relating analytic torsion with the Reidemeister torsion of the natural compactification by a manifold with boundary.