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 τ (M/B;F) by Igusa-Klein and Τ (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 τ (M/B;F) and Τ (M/B;F) in the case that there exists a function f: M → ℝ that is Morse on every fibre of p.