Morse theory and higher torsion invariants

Let p: M B be a family of compact manifolds, and let

F M

be a flat vector bundle. The higher torsion invariants (M/B;F) by Igusa-Klein and

(M/B;F)

by Bismut-Lott are both generalisations of the classical Franz-Reidemeister torsion. These invariants detect

homeomorphic, but not diffeomorphic bundles with the same given fibre X and base B, e.g. if

X is an odd-dimensional spere.

We establish a relation between (M/B;F) and (M/B;F) in the case that there exists a function

f: M that is Morse on every fibre of p.