Analytic surgery and gluing of the Bismut-Lott torsion form and eta form

Abstract

Given a fiber bundle with closed connected fibers, and a family of separating hypersurfaces, we study the behavior of the Bismut-Lott analytic torsion form, and the eta form for a duality bundle, under analytic surgery in the sense of Hassell, Mazzeo and Melrose. We find that under the surgery limit, the rescaled heat kernel is non-singular, while both the Bismut-Lott analytic torsion form and eta form can be written as the sum of a logarithmic term, which satisfies the Igusa additivity property, the b- Bismut-Lott analytic torsion form (respectively the b- eta form), and an error term coming from the reduced normal operator. Hence we obtain a gluing formula for these invariants.

arXiv commentThere are some major errors in the paper, like clarifying the limit of the torsion and eta forms at the size of the underlying manifold going to infinity