Algebraic & Geometric Topology

On symplectic fillings

John B Etnyre

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Abstract

In this note we make several observations concerning symplectic fillings. In particular we show that a (strongly or weakly) semi-fillable contact structure is fillable and any filling embeds as a symplectic domain in a closed symplectic manifold. We also relate properties of the open book decomposition of a contact manifold to its possible fillings. These results are also useful in proving property P for knots [P Kronheimer and T Mrowka, Geometry and Topology, 8 (2004) 295–310] and in showing the contact Heegaard Floer invariant of a fillable contact structure does not vanish [P Ozsvath and Z Szabo, Geometry and Topology, 8 (2004) 311–334].

Article information

Source
Algebr. Geom. Topol., Volume 4, Number 1 (2004), 73-80.

Dates
Received: 7 January 2004
Accepted: 19 January 2004
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513882467

Digital Object Identifier
doi:10.2140/agt.2004.4.73

Mathematical Reviews number (MathSciNet)
MR2023278

Zentralblatt MATH identifier
1078.53074

Subjects
Primary: 53D05: Symplectic manifolds, general 53D10: Contact manifolds, general
Secondary: 57M50: Geometric structures on low-dimensional manifolds

Keywords
tight symplectic filling convexity

Citation

Etnyre, John B. On symplectic fillings. Algebr. Geom. Topol. 4 (2004), no. 1, 73--80. doi:10.2140/agt.2004.4.73. https://projecteuclid.org/euclid.agt/1513882467


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References

  • S Akbulut, B Ozbagci, On the topology of compact Stein surfaces, Int. Math. Res. Not. 2002, no. 15, 769–782
    Zentralblatt MATH: 1007.57023
    Digital Object Identifier: doi:10.1155/S1073792802108105
  • F Ding, H Geiges, Symplectic fillability of tight contact structures on torus bundles, \agtref120018153172
    Zentralblatt MATH: 0974.53061
    Digital Object Identifier: doi:10.2140/agt.2001.1.153
  • Y Eliashberg, A few remarks about symplectic filling, \gtref820046277293
  • Y Eliashberg, Unique holomorphically fillable contact structure on the $3$-torus, Internat. Math. Res. Notices 2 (1996) 77–82
  • Y Eliashberg, Contact 3-manifolds twenty years since J Martinet's work, Ann. Inst. Fourier 42 (1992) 165–192
    Mathematical Reviews (MathSciNet): MR1162559
    Zentralblatt MATH: 0756.53017
    Digital Object Identifier: doi:10.5802/aif.1288
  • Y Eliashberg, On symplectic manifolds with some contact properties, J. Diff. Geom. 33 (1991) 233–238
    Zentralblatt MATH: 0735.53021
    Digital Object Identifier: doi:10.4310/jdg/1214446036
  • Y Eliashberg, Topological characterization of Stein manifolds of dimension $>$ 2, Int. J. of Math. 1 (1990) 29–46
  • Y Eliashberg, Filling by holomorphic discs and its applications, from: “Geometry of low-dimensional manifolds, Vol. II” (Donaldson and Thomas, editors) Cambridge (1990)
    Zentralblatt MATH: 0731.53036
  • Y Eliashberg, W Thurston, Confoliations, Amer. Math. Soc. Providence (1998)
  • J Etnyre, Symplectic convexity in low-dimensional topology, Top. Appl. 88 (1998) 3–25
    Zentralblatt MATH: 0930.53049
    Digital Object Identifier: doi:10.1016/S0166-8641(97)00196-X
  • J Etnyre, K Honda, On Symplectic Cobordisms, Math. Annalen 323 (2002) 31–39
    Zentralblatt MATH: 1022.53059
    Digital Object Identifier: doi:10.1007/s002080100292
  • J Etnyre, K Honda, Tight contact structures with no symplectic fillings, Invent. Math 148 (2002) 609–626
    Zentralblatt MATH: 1037.57020
    Digital Object Identifier: doi:10.1007/s002220100204
  • D Gay, Explicit concave fillings of contact three-manifolds, Proc. Cam. Phil. Soc. 133 (2002) 431–441
    Zentralblatt MATH: 1012.57039
  • E Giroux, Géométrie de contact: de la dimension trois vers les simensions supérieures, Proc. ICM-Beijing, 2 (2002) 405–414
  • R Gompf, Handlebody construction of Stein surfaces, Annals of Math. 148 (1998) 619–693
    Zentralblatt MATH: 0919.57012
  • N Goodman, Contact Structures and Open Books, PhD. Thesis, University of Texas Austin 2003
  • M Gromov, Pseudo-holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307–347
    Zentralblatt MATH: 0592.53025
    Digital Object Identifier: doi:10.1007/BF01388806
  • P Kronheimer, T Mrowka, Witten's conjecture and property P, \gtref820047295310
    Zentralblatt MATH: 1072.57005
    Digital Object Identifier: doi:10.2140/gt.2004.8.295
  • P Kronheimer, T Mrowka, Monopoles and contact structures, Invent. math. 130 (1997) 209–255
    Zentralblatt MATH: 0892.53015
    Digital Object Identifier: doi:10.1007/s002220050183
  • P Kronheimer, T Mrowka, P Ozsváth, Z Szabó, Monopoles and lens space surgeries.
    arXiv: math.GT/0310164
  • P Lisca, Symplectic fillings and positive scalar curvature, \gtref219986103116
    Zentralblatt MATH: 0942.53050
    Digital Object Identifier: doi:10.2140/gt.1998.2.103
  • P Lisca, On symplectic fillings of $3$-manifolds, from: “ Proceedings of 6th Gökova Geometry-Topology Conference”, Turkish J. Math. 23 (1999) 151–159
  • P Lisca, G Matić, Tight contact structures and Seiberg-Witten invariants, Invent. Math. 129 (1997) 509–525
    Mathematical Reviews (MathSciNet): MR1465333
    Zentralblatt MATH: 0882.57008
    Digital Object Identifier: doi:10.1007/s002220050171
  • P Lisca, A Stipsicz, An infinite family of tight, not semi-fillable contact three-manifolds, \gtref720033010551073
    Zentralblatt MATH: 1127.57302
    Digital Object Identifier: doi:10.2140/gt.2003.7.1055
  • D McDuff, Symplectic manifolds with contact type boundary, Invent. Math. 103 (1991) 651–671
    Mathematical Reviews (MathSciNet): MR1091622
    Zentralblatt MATH: 0719.53015
    Digital Object Identifier: doi:10.1007/BF01239530
  • D McDuff, The structure of rational and ruled symplectic $4$-manifolds, 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
    Zentralblatt MATH: 0957.57022
    Digital Object Identifier: doi:10.1007/s000140050106
  • P Ozsváth, Z Szabó, Holomorphic disks and genus bounds, \gtref820048311334
  • A Stipsicz, On the geography of Stein fillings of certain 3-manifolds, Michigan Math. J. 51 (2003) 327–337
    Zentralblatt MATH: 1043.53066
    Digital Object Identifier: doi:10.1307/mmj/1060013199
  • W P Thurston, H Winkelnkemper, On the existence of contact forms, Proc. Amer. Math. Soc. 52 (1975) 345–347
    Zentralblatt MATH: 0312.53028
    Digital Object Identifier: doi:10.1090/S0002-9939-1975-0375366-7
  • A Weinstein, Contact surgery and symplectic handlebodies, Hokkiado Math. Journal 20 (1991) 241–251
    Zentralblatt MATH: 0737.57012
    Digital Object Identifier: doi:10.14492/hokmj/1381413841

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