## Journal of Differential Geometry

### Non-exact symplectic cobordisms between contact 3-manifolds

Chris Wendl

#### 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.

#### Article information

Source
J. Differential Geom., Volume 95, Number 1 (2013), 121-182.

Dates
First available in Project Euclid: 29 July 2013

https://projecteuclid.org/euclid.jdg/1375124611

Digital Object Identifier
doi:10.4310/jdg/1375124611

Mathematical Reviews number (MathSciNet)
MR3128981

Zentralblatt MATH identifier
1278.57037

#### Citation

Wendl, Chris. Non-exact symplectic cobordisms between contact 3-manifolds. J. Differential Geom. 95 (2013), no. 1, 121--182. doi:10.4310/jdg/1375124611. https://projecteuclid.org/euclid.jdg/1375124611