A note on exotic families of 4-manifolds

TL;DR

This paper constructs examples of smooth fiber bundles over the circle with identical total spaces and topological structures but differing smooth structures, including a non-simply-connected fiber, highlighting subtle distinctions in 4-manifold topology.

Contribution

It provides explicit examples demonstrating the existence of smooth fiber bundles that are topologically but not smoothly isomorphic, with a non-simply-connected fiber.

Findings

Total spaces are diffeomorphic for the constructed bundles.

Bundles are topologically isomorphic but not smoothly isomorphic.

Includes an example with a non-simply-connected fiber.

Abstract

We present a pair of smooth fiber bundles over the circle with a common $4$-dimensional fiber with the following properties: (1) their total spaces are diffeomorphic to each other; (2) they are isomorphic to each other as topological fiber bundles; (3) they are not isomorphic to each other as smooth fiber bundles. In particular, we exhibit an example with non-simply-connected fiber.