Morse theory and higher torsion invariants II

TL;DR

This paper extends the theory of higher torsion invariants to cases with parallel metrics and Morse functions, providing new formulas and generalizations in the context of fiber bundles with flat vector bundles.

Contribution

It generalizes Igusa's higher Franz-Reidemeister torsion to include fiber-wise cohomology with parallel metrics and computes differences with higher analytic torsion under Morse conditions.

Findings

Generalization of higher torsion invariants to parallel metric cases

Explicit computation of torsion differences with analytic torsion

Extension of previous examples to broader settings

Abstract

Let p: M -> B be a family of compact manifolds equipped with a unitarily flat vector bundle F -> M. We generalize Igusa's higher Franz-Reidemeister torsion \tau(M/B;F) to the case that the fibre-wise cohomology H^*(M/B;F) -> B carries a parallel metric. If moreover M admits a fibre-wise Morse function, we compute the difference of \tau(M/B;F) and the higher analytic torsion \Cal T(M/B;F). We also generalise the examples given in math.DG/0111222 .