Small exotic Stein manifolds (Q983904)

The main result (Theorem 1.1) of this paper is the construction, for any Betti number \(b_2 \geq 1\), of pairs of homeomorphic simply connected compact Stein \(4\)-manifolds which are not diffeomorphic. Examples of this sort were previously known [\textit{A. Akhmedov, J. B. Etnyre, T. E. Mark} and \textit{I. Smith}, Math. Res. Lett. 15, No. 5--6, 1127--1132 (2008; Zbl 1156.57019)], but had large \(b_2\). Related constructions also provide the following results: Theorem 1.2. For every \(n\geq 0\) there are pairs of homeomorphic, non-diffeomorphic simply connected compact Stein 4-manifolds each with the first homology group of their boundary \(\mathbb Z/n\mathbb Z\); Theorem 1.3. There are pairs of simply connected compact Stein 4-manifolds with diffeomorphic boundaries, matching integral homology groups, but different intersection forms; and Theorem 1.4. For every \(n\geq 0\) there are pairs of homeomorphic, non-diffeomorphic simply connected compact Stein \(4\)-manifolds \(X_n\) and \(Y_n\) with \(H_2(X_n,{\mathbb Z})= H_2(Y_n,{\mathbb Z})={\mathbb Z}\), with a difference between \(X_n\) and \(Y_n\) in minimal genus for a generator of \(H_2\) of at least \(n\) . The proofs make use of techniques pioneered by the authors in [\textit{S. Akbulut} and \textit{K. Yasui}, J. Gökova Geom. Topol. GGT 2, 40--82 (2008; Zbl 1214.57027)], using Kirby calculus to enlarge corks and plugs. Consequently all constructions are explicit, and concrete handlebody pictures of their examples are given.