Non-Exact Symplectic Cobordisms Between Contact 3-Manifolds

TL;DR

This paper demonstrates that the category of contact 3-manifolds admits a rich and flexible structure of symplectic cobordisms beyond the exact case, revealing new relationships and simplifying proofs in contact topology.

Contribution

It introduces new classes of contact 3-manifolds that are symplectically cobordant to overtwisted or tight spheres, using generalized symplectic handles, and explores implications for ECH and J-holomorphic foliations.

Findings

Large classes of contact 3-manifolds are symplectically cobordant to overtwisted or tight spheres.

Construction of symplectic cobordisms using generalized handles with symplectic surfaces.

Simplified proofs of results related to fillability, planarity, and contact embeddings.

Abstract

We show that the pre-order defined on the category of contact manifolds by arbitrary symplectic cobordisms is considerably less rigid than its counterparts for exact or Stein cobordisms: in particular, we exhibit large new classes of contact 3-manifolds which are symplectically cobordant to something overtwisted, or to the tight 3-sphere, or which admit symplectic caps containing symplectically embedded spheres with vanishing self-intersection. These constructions imply new and simplified proofs of several recent results involving fillability, planarity and non-separating contact type embeddings. The cobordisms are built from generalized symplectic handles which have cores that are arbitrary symplectic surfaces with boundary and co-cores that are symplectic disks or annuli; these can be attached to contact 3-manifolds along sufficiently large neighborhoods of transverse links or pre-Lagrangian tori. We also sketch a construction of J-holomorphic foliations in these cobordisms and formulate a conjecture regarding maps induced on Embedded Contact Homology with twisted coefficients.