On Non-Separating Contact Hypersurfaces in Symplectic 4-Manifolds

TL;DR

This paper demonstrates that certain contact 3-manifolds, such as those with Giroux torsion or partial planarity, cannot be embedded as non-separating contact hypersurfaces in any closed symplectic 4-manifold, revealing restrictions on such embeddings.

Contribution

It establishes new obstructions to non-separating contact hypersurfaces in symplectic 4-manifolds, especially for manifolds with Giroux torsion or partial planarity.

Findings

Contact manifolds with Giroux torsion do not admit contact type embeddings into closed symplectic 4-manifolds.

Partially planar contact manifolds cannot be embedded as non-separating contact hypersurfaces.

Some symplectic 4-manifolds, like ruled surfaces, admit non-separating hypersurfaces that are not of contact type.

Abstract

We show that certain classes of contact 3-manifolds do not admit non-separating contact type embeddings into any closed symplectic 4-manifolds, e.g. 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-manifolds. Similarly, there are symplectic 4-manifolds that can admit smoothly embedded non-separating hypersurfaces, but not of contact type: we observe that this is the case for all symplectic ruled surfaces.