  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.

Differential Geometry