Geometry & Topology

Four-dimensional symplectic cobordisms containing three-handles

David T Gay

Full-text: Open access

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.

Article information

Source
Geom. Topol., Volume 10, Number 3 (2006), 1749-1759.

Dates
Received: 22 June 2006
Accepted: 13 October 2006
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513799778

Digital Object Identifier
doi:10.2140/gt.2006.10.1749

Mathematical Reviews number (MathSciNet)
MR2284049

Zentralblatt MATH identifier
1129.53061

Subjects
Primary: 57R17: Symplectic and contact topology 53D35: Global theory of symplectic and contact manifolds [See also 57Rxx]
Secondary: 57M50: Geometric structures on low-dimensional manifolds 53D20: Momentum maps; symplectic reduction

Keywords
symplectic cobordism contact structure 3-manifold 4-manifold 3-handle fillable Weinstein conjecture overtwisted torsion toroidal manifold moment map toric fibration

Citation

Gay, David T. Four-dimensional symplectic cobordisms containing three-handles. Geom. Topol. 10 (2006), no. 3, 1749--1759. doi:10.2140/gt.2006.10.1749. https://projecteuclid.org/euclid.gt/1513799778


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