Four-dimensional symplectic cobordisms containing three-handles

Gay, David T

doi:10.2140/gt.2006.10.1749

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Four-dimensional symplectic cobordisms containing three-handles

David T Gay

Geometry & Topology 10 (2006) 1749–1759

DOI: 10.2140/gt.2006.10.1749

arXiv: math.GT/0606402

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.

Keywords

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

Mathematical Subject Classification 2000

Primary: 57R17, 53D35

Secondary: 57M50, 53D20

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Publication

Received: 22 June 2006

Accepted: 13 October 2006

Published: 28 October 2006

Proposed: Yasha Eliashberg

Seconded: Peter Ozsváth, Tomasz Mrowka

Authors

David T Gay
Department of Mathematics and Applied Mathematics University of Cape Town Private Bag X3 Rondebosch 7701 South Africa

Volume 10, issue 3 (2006)