Exotic smooth structures on topological fibre bundles I

TL;DR

This paper develops a theory to measure differences in smooth structures on topological fiber bundles using homology classes, and relates these differences to higher Reidemeister torsion invariants, providing a comprehensive framework for understanding exotic smooth structures.

Contribution

It offers a complete, self-contained exposition of the theory linking smooth structure differences to homology classes and torsion invariants, extending previous results.

Findings

Homology class measures fiberwise tangential homeomorphism differences.

Higher Reidemeister torsion equals the Poincaré dual of the smooth structure class.

Rational and stable invariants fully characterize the differences.

Abstract

When two smooth manifold bundles over the same base are fiberwise tangentially homeomorphic, the difference is measured by a homology class in the total space of the bundle. We call this the relative smooth structure class. Rationally and stably, this is a complete invariant. We give a more or less complete and self-contained exposition of this theory which is a reformulation of some of the results of [7]. An important application is the computation of the Igusa-Klein higher Reidemeister torsion invariants of these exotic smooth structures. Namely, the higher torsion invariant is equal to the Poincar\'e dual of the image of the smooth structure class in the homology of the base. This is proved in the companion paper [11] written by the first two authors.