A gluing formula for the analytic torsion on singular spaces

TL;DR

This paper establishes a gluing formula for the analytic torsion on singular, non-compact Riemannian manifolds, enabling the extension of Cheeger-Müller type theorems to such spaces.

Contribution

It introduces a new gluing formula for analytic torsion on singular spaces, generalizing previous results to non-compact manifolds with singularities.

Findings

Proves a gluing formula for analytic torsion on singular manifolds.

Provides a criterion for the existence of heat expansion and analytic torsion.

Lays groundwork for Cheeger-Müller type theorems on singular spaces.

Abstract

We prove a gluing formula for the analytic torsion on non-compact (i.e. singular) riemannian manifolds. Let M= U\cup M_1, where M_1 is a compact manifold with boundary and U represents a model of the singularity. For general elliptic operators we formulate a criterion, which can be checked solely on U, for the existence of a global heat expansion, in particular for the existence of the analytic torsion in case of the Laplace operator. The main result then is the gluing formula for the analytic torsion. Here, decompositions M=M_1\cup_W M_2 along any compact closed hypersurface W with M_1, M_2 both non-compact are allowed; however product structure near W is assumed. We work with the de Rham complex coupled to an arbitrary flat bundle F; the metric on F is not assumed to be flat. In an appendix the corresponding algebraic gluing formula is proved. As a consequence we obtain a framework for proving a Cheeger-M\"uller type Theorem for singular manifolds; the latter has been the main motivation for this work. The main tool is Vishik's theory of moving boundary value problems for the de Rham complex which has also been successfully applied to Dirac type operators and the eta invariant by J. Br\"uning and the author. The paper also serves as a new, self--contained, and brief approach to Vishik's important work.