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 R 4 . Using an obstruction derived from Heegaard Floer homology we prove that no Brieskorn homology sphere admits a contact type embedding in R 4 , a result that has bearing on conjectures of Gompf and Kollár. This implies in particular that no rationally convex domain in C 2 has boundary diffeomorphic to a Brieskorn sphere. We also give infinitely many examples of contact 3-manifolds that bound Stein domains but not symplectically convex ones; in particular we find Stein domains in C 2 that cannot be made Weinstein with respect to the ambient symplectic structure while preserving the contact structure on their boundaries.