Vol. 6, No. 1, 2013

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
A gluing formula for the analytic torsion on singular spaces

Matthias Lesch

Vol. 6 (2013), No. 1, 221–256
Abstract

We prove a gluing formula for the analytic torsion on noncompact (i.e., singular) Riemannian manifolds. Let M = U M1M1, where M1 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 = M1 Y M2 along any compact closed hypersurface Y with M1, M2 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.

Keywords
analytic torsion, determinants
Mathematical Subject Classification 2010
Primary: 58J52
Secondary: 58J05, 58J10, 58J35
Milestones
Received: 8 June 2012
Revised: 10 September 2012
Accepted: 18 October 2012
Published: 1 June 2013
Authors
Matthias Lesch
Mathematisches Institut
Universität Bonn
Endenicher Allee 60
D-53115 Bonn
Germany
http://www.matthiaslesch.de