Duality and vanishing theorems for topologically trivial families of smooth h-cobordisms

TL;DR

This paper introduces a new invariant called the smooth structure class for topologically trivial smooth h-cobordism families, linking it to higher torsion and proving a duality theorem that supports the Rigidity Conjecture.

Contribution

It defines the smooth structure class, computes it via fiberwise Morse theory, and establishes a duality theorem leading to a vanishing result confirming the Rigidity Conjecture.

Findings

The smooth structure class relates to higher Franz–Reidemeister torsion.

A duality theorem generalizes Milnor's duality for Whitehead torsion.

A vanishing theorem supports the Rigidity Conjecture for manifold bundles.

Abstract

Using the work of Dwyer, Weiss, and Williams we associate an invariant to any topologically trivial family of smooth h-cobordisms. This invariant is called the smooth structure class, and is closely related to the higher Franz--Reidemeister torsion of Igusa and Klein. We compute the smooth structure class in terms of a fiberwise generalized Morse function using fiberwise Poincar\'e--Hopf theory. This computation gives rise to a duality theorem for the smooth structure class that generalizes Milnor's duality theorem for the Whitehead torsion. From this result we deduce a vanishing theorem that implies the Rigidity Conjecture of Goette and Igusa. This conjecture states that, after rationalizing, there are no stable exotic smoothings of manifold bundles with closed even dimensional fibers.