Decomposition of the {$\zeta$}-determinant for the Laplacian on manifolds with cylindrical end

Abstract

In this paper we combine elements of the b-calculus and elliptic boundary problems to solve the decomposition problem for the (regularized) �-determinant of the Laplacian on a manifold with cylin- drical end into the �-determinants of the Laplacians with Dirichlet con- ditions on the manifold with boundary and on the half infinite cylinder. We also compute all the contributions to this formula explicitly. We investigate the 'Mayer-Vietoris' or 'cut and paste' decomposition for- mula of the �-determinant for a Laplacian on a manifold with cylindrical end into the �-determinants of the Laplacians with Dirichlet conditions on the manifold with boundary and on the half infinite cylinder. We also introduce a new method to attack such surgery problems by comparing the problem to a corresponding model problem. This approach works for compact manifolds as well as manifolds with cylindrical ends and will be used to solve related decomposition problems for the spectral invariants of Dirac type operators in (12), (13). We remark that the noncompactness of the underlying manifold introduces many new facets and obstacles not found in the compact case, as we will explain later. We begin with a brief account of zeta determinants. The �-determinant of a Laplace type operator was pioneered in the seminal paper (21) by Ray and Singer. They were seeking an analytic version of the so- called Reidemeister torsion, a combinatorial-topological invariant introduced by Reidemeister (22) and Franz (6). They conjectured that their analytic in- variant was the same as the Reidemeister torsion. Later, this conjecture was proved independently by Cheeger (4) and Muller (17). The �-determinants have also been of great use in quantum field theory where they are being used to develop rigorous models for Feynman path integral techniques (10). Because of their use in differential topology and quantum field theory, much work has been done on understanding the nature of�-determinants, especially their behavior under 'cutting and pasting' of manifolds. This was initiated