Journal of Symplectic Geometry
Volume 15 (2017)
Nonexistence of Stein structures on 4-manifolds and maximal Thurston–Bennequin numbers
Pages: 91 – 105
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 boundary $S^3$ admitting no Stein structure.