Non-exact symplectic cobordisms between contact 3-manifolds

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 symplectic handles of the form $\Sigma \times \mathbb{D}$ and $\Sigma \times [−1, 1] \times S^1$, which have symplectic cores and can be attached to contact 3-manifolds along sufficiently large neighborhoods of transverse links and 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.