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

Kouichi Yasui

Published 2015-08-06Version 1

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 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.