Geometry & Topology

Four-dimensional symplectic cobordisms containing three-handles

David T Gay

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R17: Symplectic and contact topology 53D35: Global theory of symplectic and contact manifolds [See also 57Rxx]
Secondary: 57M50: Geometric structures on low-dimensional manifolds 53D20: Momentum maps; symplectic reduction

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


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.

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  • 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