Volume 10, issue 3 (2006)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 21
Issue 5, 2557–3190
Issue 4, 1931–2555
Issue 3, 1285–1930
Issue 2, 647–1283
Issue 1, 1–645

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Author Index
To Appear
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