On contact type hypersurfaces in 4-space

Abstract

We consider constraints on the topology of closed 3-manifolds that can arise as hypersurfaces of contact type in standard symplectic \({\mathbb {R}}^4\). Using an obstruction derived from Heegaard Floer homology we prove that no Brieskorn homology sphere admits a contact type embedding in \({\mathbb {R}}^4\), a result that has bearing on conjectures of Gompf and Kollár. This implies in particular that no rationally convex domain in \({\mathbb {C}}^2\) has boundary diffeomorphic to a Brieskorn sphere. We also give infinitely many examples of contact 3-manifolds that bound Stein domains in \({\mathbb {C}}^2\) but not domains that are symplectically convex with respect to the standard symplectic structure; in particular we find Stein domains in \({\mathbb {C}}^2\) that cannot be made Weinstein with respect to the ambient symplectic structure while preserving the contact structure on their boundaries. Finally, we observe that any strictly pseudoconvex, polynomially convex domain in \({\mathbb {C}}^2\) having rational homology sphere boundary is diffeomorphic to the standard 4-ball.