## Geometry & Topology

### Four-dimensional symplectic cobordisms containing three-handles

David T Gay

#### Abstract

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.

#### Article information

Source
Geom. Topol., Volume 10, Number 3 (2006), 1749-1759.

Dates
Received: 22 June 2006
Accepted: 13 October 2006
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513799778

Digital Object Identifier
doi:10.2140/gt.2006.10.1749

Mathematical Reviews number (MathSciNet)
MR2284049

Zentralblatt MATH identifier
1129.53061

#### Citation

Gay, David T. Four-dimensional symplectic cobordisms containing three-handles. Geom. Topol. 10 (2006), no. 3, 1749--1759. doi:10.2140/gt.2006.10.1749. https://projecteuclid.org/euclid.gt/1513799778

#### References

• F Bourgeois, V Colin, Homologie de contact des variétés toroï dales, Geom. Topol. 9 (2005) 299–313
• F Bourgeois, Y Eliashberg, H Hofer, K Wysocki, E Zehnder, Compactness results in symplectic field theory, Geom. Topol. 7 (2003) 799–888
• V Colin, Une infinité de structures de contact tendues sur les variétés toroï dales, Comment. Math. Helv. 76 (2001) 353–372
• F Ding, H Geiges, Symplectic fillability of tight contact structures on torus bundles, Algebr. Geom. Topol. 1 (2001) 153–172
• Y Eliashberg, Classification of overtwisted contact structures on $3$-manifolds, Invent. Math. 98 (1989) 623–637
• Y Eliashberg, Filling by holomorphic discs and its applications, from: “Geometry of low-dimensional manifolds, 2 (Durham, 1989)”, LMS Lecture Notes 151, Cambridge Univ. Press, Cambridge (1990) 45–67
• Y Eliashberg, Topological characterization of Stein manifolds of dimension $>2$, Internat. J. Math. 1 (1990) 29–46
• Y Eliashberg, Unique holomorphically fillable contact structure on the $3$-torus, Internat. Math. Res. Notices (1996) 77–82
• Y Eliashberg, A few remarks about symplectic filling, Geom. Topol. 8 (2004) 277–293
• J B Etnyre, On symplectic fillings, Algebr. Geom. Topol. 4 (2004) 73–80
• J B Etnyre, K Honda, On symplectic cobordisms, Math. Ann. 323 (2002) 31–39
• D T Gay, Explicit concave fillings of contact three-manifolds, Math. Proc. Cambridge Philos. Soc. 133 (2002) 431–441
• H Geiges, Contact geometry, from: “Handbook of differential geometry. Vol. II”, Elsevier/North-Holland, Amsterdam (2006) 315–382
• P Ghiggini, Ozsváth-Szabó invariants and fillability of contact structures, Math. Z. 253 (2006) 159–175
• E Giroux, Une structure de contact, même tendue, est plus ou moins tordue, Ann. Sci. École Norm. Sup. $(4)$ 27 (1994) 697–705
• E Giroux, Une infinité de structures de contact tendues sur une infinité de variétés, Invent. Math. 135 (1999) 789–802
• E Giroux, Structures de contact en dimension trois et bifurcations des feuilletages de surfaces, Invent. Math. 141 (2000) 615–689
• M Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307–347
• P Lisca, A I Stipsicz, Symplectic fillability and Giroux torsion
• D McDuff, }, J. Amer. Math. Soc. 3 (1990) 679–712
• H Ohta, K Ono, Simple singularities and topology of symplectically filling $4$-manifold, Comment. Math. Helv. 74 (1999) 575–590
• A I Stipsicz, On the geography of Stein fillings of certain 3-manifolds, Michigan Math. J. 51 (2003) 327–337
• M Symington, Four dimensions from two in symplectic topology, from: “Topology and geometry of manifolds (Athens, GA, 2001)”, Proc. Sympos. Pure Math. 71, Amer. Math. Soc., Providence, RI (2003) 153–208
• C H Taubes, The Seiberg-Witten invariants and symplectic forms, Math. Res. Lett. 1 (1994) 809–822
• A Weinstein, Contact surgery and symplectic handlebodies, Hokkaido Math. J. 20 (1991) 241–251