Four-dimensional symplectic cobordisms containing three-handles

By David T. Gay. 2006

60 citations recorded by the source corpus.

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 twospheres 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.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d7e2a329d530c58b832802dd958d22be17c6fe98

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Citations in the corpus

Weak Symplectic Fillings and Holomorphic Curves (2010, unknown): “Theorem (Gay [Gay06] and Ghiggini-Honda [GH08]).”
The Seiberg–Witten equations and the Weinstein conjecture (2006, result): “The most recent results known to the author are those of Etnyre and Gay [8], Colin and Honda [6] and Abbas, Cielebak and Hofer [1].”
Algebraic Torsion in Contact Manifolds (2010, background): “It was shown by D. Gay [Gay06] that contact 3-manifolds with Giroux torsion are never strongly fillable, and a computation of the twisted Ozsváth-Szabó contact invariant due to Ghiggini and Honda [GH] shows that Giroux torsion is also an obstruction to weak fillings whenever the submanifold T 2 ×…”
Weak and strong fillability of higher dimensional contact manifolds (2011, background): “This invariant, introduced by Giroux in [Gir00], measures the amount that a contact structure “twists” in neighborhoods of certain embedded 2-tori; it does not imply overtwistedness but does obstruct strong [Gay06] and sometimes also weak [GH08] fillability.”
Strongly fillable contact manifolds and $J$-holomorphic foliations (2008, background): “Note that an easier proof that strongly fillable manifolds are tight is possible using the result for Giroux torsion below; cf. [Gay06, Corollary 5].”
Lectures on Holomorphic Curves in Symplectic and Contact Geometry (2010, background): “Theorem 1.54 (D. Gay [Gay06], P. Ghiggini and K. Honda [GH]).”
A hierarchy of local symplectic filling obstructions for contact $3$-manifolds (2010, background): “Contact manifolds containing such an object are said to have Giroux torsion, and the fact that they are not fillable in general is a comparatively recent result, due to Gay [Gay06].”