Contact 5-manifolds and smooth structures on Stein 4-manifolds

TL;DR

This paper demonstrates how contact 5-manifolds can distinguish smooth structures on Stein 4-manifolds using contact open books, providing conditions for infinite families of non-diffeomorphic Stein 4-manifolds.

Contribution

It introduces a new method to differentiate smooth structures on Stein 4-manifolds via contact 5-manifolds and contact open books, with criteria for infinite non-diffeomorphic families.

Findings

Contact 5-manifolds can distinguish smooth structures on Stein 4-manifolds.

A simple condition ensures infinite non-diffeomorphic Stein 4-manifolds.

Examples of infinitely many exotic Stein 4-manifolds distinguished by these methods.

Abstract

We show that, under a certain condition, contact 5-manifolds can `coarsely' distinguish smooth structures on compact Stein 4-manifolds via contact open books. We also give a simple sufficient condition for an infinite family of Stein 4-manifolds to have an infinite subfamily of pairwise non-diffeomorphic Stein 4-manifolds. The proofs rely on the adjunction inequality. We remark that there are various examples of infinitely many pairwise exotic Stein 4-manifolds whose smooth structures can be distinguished by these results.