Nonexistence of Stein structures on 4-manifolds and maximal Thurston-Bennequin numbers

TL;DR

This paper investigates the relationship between Stein structures on 4-manifolds and Thurston-Bennequin numbers, showing that the known sufficient condition is not necessary or that certain contractible 4-manifolds lack Stein structures.

Contribution

It proves either the converse of the known criterion is false or such contractible 4-manifolds with Stein fillable boundary do not admit Stein structures.

Findings

The converse of the known criterion may be false.

Existence of contractible 4-manifolds with Stein fillable boundary but no Stein structure.

Implications for the existence of exotic smooth structures on S^4.

Abstract

For a 4-manifold represented by a framed knot in $S^3$, it has been well known that the 4-manifold admits a Stein structure if the framing is less than the maximal Thurston-Bennequin number of the knot. In this paper, we prove either the converse of this fact is false or there exists a compact contractible oriented smooth 4-manifold (with Stein fillable boundary) admitting no Stein structure. Note that an exotic smooth structure on $S^4$ exists if and only if there exists a compact contractible oriented smooth 4-manifold with $S^3$ boundary admitting no Stein structure.