Four-dimensional symplectic cobordisms containing three-handles

Gay, David T¹

¹ Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag X3, Rondebosch 7701, South Africa

Geometry & topology, Tome 10 (2006) no. 3, pp. 1749-1759.

Résumé

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.

Détail

DOI : 10.2140/gt.2006.10.1749

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

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