Fourdimensional symplectic cobordisms containing threehandles
Abstract
We construct fourdimensional symplectic cobordisms between contact threemanifolds generalizing an example of Eliashberg. One key feature is that any handlebody decomposition of one of these cobordisms must involve threehandles. The other key feature is that these cobordisms contain chains of symplectically embedded twospheres of square zero. This, together with standard gauge theory, is used to show that any contact threemanifold of nonzero 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 pseudoholomorphic curves implies that any contact threemanifold of nonzero torsion satisfies the Weinstein conjecture. We also get examples of weakly symplectically fillable contact threemanifolds which are (strongly) symplectically cobordant to overtwisted contact threemanifolds, shedding new light on the structure of the set of contact threemanifolds equipped with the strong symplectic cobordism partial order.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2006
 arXiv:
 arXiv:math/0606402
 Bibcode:
 2006math......6402G
 Keywords:

 Mathematics  Geometric Topology;
 Mathematics  Symplectic Geometry;
 53D35;
 57R17;
 53D20;
 57M50
 EPrint:
 This is the version published by Geometry &