On nonseparating contact hypersurfaces in symplectic $4$–manifolds

Abstract

We show that certain classes of contact $3$–manifolds do not admit nonseparating contact type embeddings into any closed symplectic $4$–manifold, eg this is the case for all contact manifolds that are (partially) planar or have Giroux torsion. The latter implies that manifolds with Giroux torsion do not admit contact type embeddings into any closed symplectic $4$–manifold. Similarly, there are symplectic $4$–manifolds that can admit smoothly embedded nonseparating hypersurfaces, but not of contact type: we observe that this is the case for all symplectic ruled surfaces.