Some remarks on $h$-cobordisms between smooth 4-manifolds

TL;DR

This paper explores conditions under which the smooth $h$-cobordism theorem might hold or fail in 4-manifolds, addressing open questions in differential topology.

Contribution

It provides new criteria for when smooth $h$-cobordisms imply $s$-cobordisms and identifies limitations of standard construction methods.

Findings

New conditions ensuring positive answers to open questions

Criteria under which standard $h$-cobordism constructions fail

Insights into the structure of smooth 4-manifolds and cobordisms

Abstract

It is not known whether the realisation part of the $s$-cobordism theorem holds for smooth 4-manifolds, nor whether every pair of smoothly $h$-cobordant 4-manifolds is also smoothly $s$-cobordant. We provide some new conditions under which these questions admit a positive answer. We also give conditions under which the `standard' method to construct an $h$-cobordism with specified torsion cannot work.