Analytic surgery and analytic torsion

Abstract

Abstract. Analytic surgery, as defined in [9] and [6], is a one-parameter metric deformation of a Riemannian manifold M , which stretches M across a separating hypersurface H in a cylindrical fashion; the singular limit is a complete manifold with asymptotically cylindrical ends, M . In this paper, the analysis of [9] and [6] is used to study the behaviour of analytic torsion of unitary representations under analytic surgery. A gluing formula is obtained relating the analytic torsion of M to the 'b-analytic torsion' b T (a regularized analytic torsion on manifolds with boundary) of M . This is then used to prove the Cheeger-Müller theorem, asserting the equality of analytic and Reidemeister torsion τ on closed manifolds, and to prove the following combinatorial formula for b-analytic torsion on odd dimensional manifolds with boundary:As a step in the proof, a Hodge-theoretic description of the Mayer-Vietoris sequence for cohomology under analytic surgery is developed.

Adiabatic decomposition of the -determinant and Dirichlet to Neumann operator

“…The formula (1.8) in Theorem 1.1 has been used in [1] where the adiabatic surgery formula of the determinant line bundle is investigated. The related decomposition formula for the analytic torsion was also worked out by Hassell in [3]. He proved the analytic surgery formula of the analytic torsion using the b-calculus.…”

Adiabatic Cobordism Theorems for Analytic Torsion andη-Invariant

“…Interesting results on the cutting and pasting of the analytic torsion were obtained recently by Andrew Hassel in his doctoral thesis [10]. He uses b-pseudodifferential calculus developed by Melrose and his collaborators.…”

“…One choice is the generalized Atiyah Patodi Singer problem. This leads to a nice theory discussed in a series of papers by the second author and his collaborators (see [8], [13], [19], [20]) and many others (see for instance [4], [6], [7], [10], [14], [16]). …”

A Cheeger–Müller theorem for Cappell–Miller torsions on manifolds with boundary

“…Now consider X with the boundary Y = ∅. In [16,28] and [14], the authors studied the Ray-Singer analytic torsion under the assumption that the Hermitian metric h F on F is flat and the Riemannian metric g TX has product structure near the boundary. Dai and Fang [9] studied the case that the Hermitian metric h F is flat but without assuming a product structure for g TX near Y .…”