Exotic smooth structures on topological fibre bundles II

TL;DR

This paper constructs nearly all stable exotic smooth structures on certain high-dimensional fiber bundles using a variation of Hatcher's method, linking these structures to higher torsion invariants via a homology class.

Contribution

It introduces a new construction method for exotic smooth structures on fiber bundles with large odd-dimensional fibers and relates these structures to higher torsion invariants through homology classes.

Findings

Constructs almost all stable exotic smooth structures on specified fiber bundles.

Establishes a correspondence between exotic structures and higher torsion classes.

Links homology classes of exotic structures to Poincaré duals of higher Igusa-Klein torsion.

Abstract

We use a variation of a classical construction of A. Hatcher to construct virtually all stable exotic smooth structures on compact smooth manifold bundles whose fibers have sufficiently large odd dimension (at least twice the base dimension plus 3). Using a variation of the Dwyer-Weiss-Williams smoothing theory which we explain in a separate joint paper with Bruce Williams [11], we associate a homology class in the total space of the bundle to each exotic smooth structure and we show that the image of this class in the homology of the base is the Poincar\'e dual of the relative higher Igusa-Klein (IK) torsion invariant. This answers the question, in the relative case, of which cohomology classes can occur as relative higher torsion classes.