Four-dimensional symplectic cobordisms containing three-handles

Abstract

We construct four-dimensional symplectic cobordisms between contact three-manifolds generalizing an example of Eliashberg. One key feature is that any handlebody decomposition of one of these cobordisms must involve three-handles. The other key feature is that these cobordisms contain chains of symplectically embedded two-spheres of square zero. This, together with standard gauge theory, is used to show that any contact three-manifold of non-zero torsion (in the sense of Giroux) cannot be strongly symplectically fillable. John Etnyre pointed out to the author that the same argument together with compactness results for pseudo-holomorphic curves implies that any contact three-manifold of non-zero torsion satisfies the Weinstein conjecture. We also get examples of weakly symplectically fillable contact three-manifolds which are (strongly) symplectically cobordant to overtwisted contact three-manifolds, shedding new light on the structure of the set of contact three-manifolds equipped with the strong symplectic cobordism partial order.

This is the version published by Geometry & Topology on 28 October 2006

References in corpus (6)

• Compactness results in Symplectic Field Theory
• Structures de contact en dimension trois et bifurcations des feuilletages de surfaces
• Symplectic fillability of tight contact structures on torus bundles
• On symplectic fillings
• Explicit concave fillings of contact three-manifolds
• Homologie de contact des varietes toroidales

Cited by in corpus (18)

• The Seiberg-Witten equations and the Weinstein conjecture
• Strongly Fillable Contact Manifolds and J-holomorphic Foliations
• Weak and strong fillability of higher dimensional contact manifolds
• Weak Symplectic Fillings and Holomorphic Curves
• A Hierarchy of Local Symplectic Filling Obstructions for Contact 3-Manifolds
• Cylindrical contact homology and topological entropy
• Algebraic Torsion in Contact Manifolds
• On Non-Separating Contact Hypersurfaces in Symplectic 4-Manifolds
• Tight small Seifert fibered manifolds with $e_0=-2$
• Canonical contact structures on some singularity links
• Symplectic Geometry of Anosov Flows in Dimension 3 and Bi-Contact Topology
• On Stein fillings of contact torus bundles
• Orbit equivalences of $\mathbb{R}$-covered Anosov flows and hyperbolic-like actions on the line
• Exactly fillable contact structures without Stein fillings
• Contact Structures on Plumbed 3-Manifolds
• Anosov flows and Liouville pairs in dimension three
• Elliptic open books on torus bundles over the circle
• A remark about weak fillings