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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Four-dimensional symplectic cobordisms containing three-handles

David T Gay

Geometry & Topology 10 (2006) 1749–1759

arXiv: math.GT/0606402


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.

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
Forward citations
Received: 22 June 2006
Accepted: 13 October 2006
Published: 28 October 2006
Proposed: Yasha Eliashberg
Seconded: Peter Ozsváth, Tomasz Mrowka
David T Gay
Department of Mathematics and Applied Mathematics
University of Cape Town
Private Bag X3
Rondebosch 7701
South Africa