A gluing formula for the analytic torsion on singular spaces

Abstract

We prove a gluing formula for the analytic torsion on noncompact (i.e., singular) Riemannian manifolds. Let $M=U{\cup }_{\partial M_1}{M}_1$, 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 the case of the Laplace operator. The main result then is the gluing formula for the analytic torsion. Here, decompositions $M=M_1{\cup }_Y{M}_2$ along any compact closed hypersurface $Y$ with $M_1$, $M_2$ both noncompact are allowed; however a product structure near $Y$ 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–Müller-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. Brüning and the author. The paper also serves as a new, self-contained, and brief approach to Vishik’s important work.