Duke Mathematical Journal

Strongly fillable contact manifolds and J-holomorphic foliations

Chris Wendl

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We prove that every strong symplectic filling of a planar contact manifold admits a symplectic Lefschetz fibration over the disk, and every strong filling of T3 similarly admits a Lefschetz fibration over the annulus. It follows that strongly fillable planar contact structures are also Stein fillable, and all strong fillings of T3 are equivalent up to symplectic deformation and blowup. These constructions result from a compactness theorem for punctured J-holomorphic curves that foliate a convex symplectic manifold. We use it also to show that the compactly supported symplectomorphism group on T*T2 is contractible, and to define an obstruction to strong fillability that yields a non-gauge-theoretic proof of Gay's recent nonfillability result [G] for contact manifolds with positive Giroux torsion

Article information

Duke Math. J., Volume 151, Number 3 (2010), 337-384.

First available in Project Euclid: 8 February 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32Q65: Pseudoholomorphic curves
Secondary: 57R17: Symplectic and contact topology


Wendl, Chris. Strongly fillable contact manifolds and $J$ -holomorphic foliations. Duke Math. J. 151 (2010), no. 3, 337--384. doi:10.1215/00127094-2010-001. https://projecteuclid.org/euclid.dmj/1265637657

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  • C. Abbas, Holomorphic open book decompositions, preprint.
  • C. Abbas, K. Cieliebak, and H. Hofer, The Weinstein conjecture for planar contact structures in dimension three, Comment. Math. Helv. 80 (2005), 771--793.
  • S. Akbulut and B. Ozbagci, Lefschetz fibrations on compact Stein surfaces, Geom. Topol. 5 (2001), 319--334.
  • A. Akhmedov, J. B. Etnyre, T. E. Mark, and I. Smith, A note on Stein fillings of contact manifolds, Math. Res. Lett. 15 (2008), 1127--1132.
  • P. Albers, B. Bramham, and C. Wendl, On non-separating contact hypersurfaces in symplectic $4$-manifolds, preprint.
  • D. Auroux, S. K. Donaldson, and L. Katzarkov, Luttinger surgery along Lagrangian tori and non-isotopy for singular symplectic plane curves, Math. Ann. 326 (2003), 185--203.
  • F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki, and E. Zehnder, Compactness results in symplectic field theory, Geom. Topol. 7 (2003), 799--888.
  • F. Bourgeois and K. Mohnke, Coherent orientations in symplectic field theory, Math. Z. 248 (2004), 123--146.
  • D. L. Dragnev, Fredholm theory and transversality for noncompact pseudoholomorphic maps in symplectizations, Comm. Pure Appl. Math. 57 (2004), 726--763.
  • Y. Eliashberg, ``Filling by holomorphic discs and its applications'' in Geometry of Low-dimensional Manifolds, 2 (Durham, England, 1989), London Math. Soc. Lecture Note Ser. 151, Cambridge Univ. Press, Cambridge, 1990, 45--67.
  • —, Topological characterization of Stein manifolds of dimension $>$ 2, Internat. J. Math. 1 (1990), 29--46.
  • —, Unique holomorphically fillable contact structure on the $3$-torus, Internat. Math. Res. Notices 1996, no. 2, 77--82.
  • J. B. Etnyre, ``Symplectic convexity in low-dimensional topology'' in Symplectic, Contact and Low-dimensional Topology (Athens, Georgia, 1996), Topology Appl. 88 (1998), 3--25.
  • —, Planar open book decompositions and contact structures, Int. Math. Res. Not. 2004, no. 79, 4255--4267.
  • —, ``Lectures on open book decompositions and contact structures'' in Floer Homology, Gauge Theory, and Low-dimensional Topology, Clay Math. Proc. 5, Amer. Math. Soc., Providence, 2006, 103--141.
  • D. T. Gay, Four-dimensional symplectic cobordisms containing three-handles, Geom. Topol. 10 (2006), 1749--1759.
  • P. Ghiggini, Strongly fillable contact $3$-manifolds without Stein fillings, Geom. Topol. 9 (2005), 1677--1687.
  • P. Ghiggini, K. Honda, and J. Van Horn-Morris, The vanishing of the contact invariant in the presence of torsion, preprint.
  • E. Giroux, Une structure de contact, même tendue, est plus ou moins tordue, Ann. Sci. École Norm. Sup. (4) 27 (1994), 697--705.
  • —, Lecture given at Georgia Topology Conference, May 24, 2001; notes available at http://www.math.uga.edu/$\sim $topology/2001/giroux.pdf
  • R. E. Gompf and A. I. Stipsicz, $4$-manifolds and Kirby calculus, Grad. Stud. Math. 20, Amer. Math. Soc., Providence, 1999.
  • M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307--347.
  • R. Hind, Stein fillings of lens spaces, Commun. Contemp. Math. 5 (2003), 967--982.
  • H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math. 114 (1993), 515--563.
  • —, ``Holomorphic curves and real three-dimensional dynamics'' in GAFA 2000 (Tel Aviv, 1999), Geom. Funct. Anal. 2000, Special Volume, Part II, 674--704.
  • H. Hofer, K. Wysocki, and E. Zehnder, Properties of pseudo-holomorphic curves in symplectisations, II: Embedding controls and algebraic invariants, Geom. Funct. Anal. 5 (1995), 270--328.
  • —, Properties of pseudoholomorphic curves in symplectisations, I: Asymptotics, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), 337--379.
  • —, ``Properties of pseudoholomorphic curves in symplectisations, IV: Asymptotics with degeneracies'' in Contact and Symplectic Geometry (Cambridge, 1994), Publ. Newton Inst. 8, Cambridge Univ. Press, Cambridge, 1996, 78--117.
  • —, Finite energy foliations of tight three-spheres and Hamiltonian dynamics, Ann. of Math. (2) 157 (2003), 125--255.
  • P. Lisca, On symplectic fillings of lens spaces, Trans. Amer. Math. Soc. 360, no. 2 (2008), 765--799.
  • A. Loi and R. Piergallini, Compact Stein surfaces with boundary as branched covers of B$^4$, Invent. Math. 143 (2001), 325--348.
  • K. M. Luttinger, Lagrangian tori in R$^4$, J. Differential Geom. 42 (1995), 220--228.
  • D. Mcduff, The structure of rational and ruled symplectic $4$-manifolds, J. Amer. Math. Soc. 3 (1990), 679--712.
  • D. Mcduff and D. Salamon, J-holomorphic curves and symplectic topology, Amer. Math. Soc. Colloq. Publ. 52, Amer. Math. Soc., Providence, 2004.
  • H. Ohta and K. Ono, Simple singularities and symplectic fillings, J. Differential Geom. 69 (2005), 1--42.
  • B. Ozbagci and A. I. Stipsicz, Surgery on Contact $3$-manifolds and Stein Surfaces, Bolyai Soc. Math. Stud. 13, Springer, Berlin, 2004.
  • R. Siefring, Relative asymptotic behavior of pseudoholomorphic half-cylinders, Comm. Pure Appl. Math. 61 (2008), 1631--1684.
  • —, Intersection theory of punctured pseudoholomorphic curves, preprint.
  • R. Siefring and C. Wendl, Pseudoholomorphic curves, intersections and Morse-Bott asymptotics, in preparation.
  • A. I. Stipsicz, Gauge theory and Stein fillings of certain $3$-manifolds, Turkish J. Math. 26 (2002), 115--130.
  • C. H. Taubes, The Seiberg-Witten equations and the Weinstein conjecture, Geom. Topol. 11 (2007), 2117--2202.
  • J. Van Horn-Morris, Constructions of open book decompositions, Ph.D. dissertation, University of Texas at Austin, Austin, Tex., 2007.
  • C. Wendl, Finite energy foliations on overtwisted contact manifolds, Geom. Topol. 12 (2008), 531--616.
  • —, Finite energy foliations and surgery on transverse links, Ph.D. dissertation, New York University, New York, N.Y., 2005.
  • —, Compactness for embedded pseudoholomorphic curves in $3$-manifolds, to appear in J. Eur. Math. Soc. (JEMS), preprint.
  • —, Automatic transversality and orbifolds of punctured holomorphic curves in dimension four, to appear in Comment. Math. Helv., preprint.
  • —, Open book decompositions and stable Hamiltonian structures, to appear in Expos. Math., preprint.
  • —, Contact fiber sums, monodromy maps and symplectic fittings, in preparation.
  • —, Punctured holomorphic curves with boundary in $3$-manifolds: Fredholm theory and embededdness, in preparation.