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**Non-exact symplectic cobordisms between contact 3-manifolds.**
*(English)*
Zbl 1278.57037

The paper under review discusses the (non-symmetric) pre-order defined by connected, symplectic cobordisms on the set of closed, connected, contact \(3\)-manifolds.

In [Duke Math. J. 162, No. 12, 2197–2283 (2013; Zbl 1279.57019)] the author had defined a notion of “planar \(k\)-torsion domains” (for integers \(k\)) in a contact manifold and thus obtained a hierarchy of obstructions for symplectic fillability, where existence of a planar \(0\)-torsion domain was equivalent to overtwistedness. In the paper under review it is shown that existence of a planar \(k\)-torsion domain (for some \(k\)) in a contact \(3\)-manifold implies the manifold to be symplectically cobordant to an overtwisted contact manifold. Together with results of Etnyre-Honda and Gay this implies then that contact manifolds with planar \(k\)-torsion are symplectically cobordant to any contact manifold. Under weaker assumptions, assuming only the existence of a partially planar domain with nonempty binding, it is shown that a contact manifold must be symplectically cobordant to the standard tight 3-sphere and thus to every connected, fillable contact manifold.

There are many other results as well as new and simplified proofs of several recent results involving fillability, planarity and non-separating contact type embeddings in the paper.

In [Duke Math. J. 162, No. 12, 2197–2283 (2013; Zbl 1279.57019)] the author had defined a notion of “planar \(k\)-torsion domains” (for integers \(k\)) in a contact manifold and thus obtained a hierarchy of obstructions for symplectic fillability, where existence of a planar \(0\)-torsion domain was equivalent to overtwistedness. In the paper under review it is shown that existence of a planar \(k\)-torsion domain (for some \(k\)) in a contact \(3\)-manifold implies the manifold to be symplectically cobordant to an overtwisted contact manifold. Together with results of Etnyre-Honda and Gay this implies then that contact manifolds with planar \(k\)-torsion are symplectically cobordant to any contact manifold. Under weaker assumptions, assuming only the existence of a partially planar domain with nonempty binding, it is shown that a contact manifold must be symplectically cobordant to the standard tight 3-sphere and thus to every connected, fillable contact manifold.

There are many other results as well as new and simplified proofs of several recent results involving fillability, planarity and non-separating contact type embeddings in the paper.

Reviewer: Thilo Kuessner (Seoul)

### MSC:

57R90 | Other types of cobordism |

57R17 | Symplectic and contact topology in high or arbitrary dimension |