Journal of Symplectic Geometry

Volume 15 (2017)

Number 1

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

Pages: 91 – 105

DOI: https://dx.doi.org/10.4310/JSG.2017.v15.n1.a3

Author

Kouichi Yasui (Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka, Japan)

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 boundary $S^3$ admitting no Stein structure.

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Published 28 April 2017