## Duke Mathematical Journal

### A hierarchy of local symplectic filling obstructions for contact $3$-manifolds

Chris Wendl

#### Abstract

We generalize the familiar notions of overtwistedness and Giroux torsion in $3$-dimensional contact manifolds, defining an infinite hierarchy of local filling obstructions called planar torsion, whose integer-valued order $k\ge0$ can be interpreted as measuring a gradation in “degrees of tightness” of contact manifolds. We show in particular that any contact manifold with planar torsion admits no contact-type embeddings into any closed symplectic $4$-manifold, and has vanishing contact invariant in embedded contact homology, and we give examples of contact manifolds that have planar $k$-torsion for any $k\ge2$ but no Giroux torsion. We also show that the complement of the binding of a supporting open book never has planar torsion. The unifying idea in the background is a decomposition of contact manifolds in terms of contact fiber sums of open books along their binding. As the technical basis of these results, we establish existence, uniqueness, and compactness theorems for certain classes of $J$-holomorphic curves in blown-up summed open books; these also imply algebraic obstructions to planarity and embeddings of partially planar domains.

#### Article information

Source
Duke Math. J., Volume 162, Number 12 (2013), 2197-2283.

Dates
First available in Project Euclid: 9 September 2013

https://projecteuclid.org/euclid.dmj/1378729687

Digital Object Identifier
doi:10.1215/00127094-2348333

Mathematical Reviews number (MathSciNet)
MR3102479

Zentralblatt MATH identifier
1279.57019

#### Citation

Wendl, Chris. A hierarchy of local symplectic filling obstructions for contact $3$ -manifolds. Duke Math. J. 162 (2013), no. 12, 2197--2283. doi:10.1215/00127094-2348333. https://projecteuclid.org/euclid.dmj/1378729687

#### References

• [A] C. Abbas, Holomorphic open book decompositions, Duke Math. J. 158 (2011), 29–82.
• [ACH] C. Abbas, K. Cieliebak, and H. Hofer, The Weinstein conjecture for planar contact structures in dimension three, Comment. Math. Helv. 80 (2005), 771–793.
• [ABW] P. Albers, B. Bramham, and C. Wendl, On nonseparating contact hypersurfaces in symplectic $4$-manifolds, Algebr. Geom. Topol. 10 (2010), 697–737.
• [BV] İ. Baykur and J. Van Horn-Morris, Families of contact $3$-manifolds with arbitrarily large Stein fillings, with an appendix by S. Lisi and C. Wendl, preprint, arXiv:1208.0528.
• [B1] F. Bourgeois, A Morse-Bott approach to contact homology, Ph.D. dissertation, Stanford University, Stanford, California, 2002.
• [B2] F. Bourgeois, Contact homology and homotopy groups of the space of contact structures, Math. Res. Lett. 13 (2006), 71–85.
• [BEH+] F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki, and E. Zehnder, Compactness results in symplectic field theory, Geom. Topol. 7 (2003), 799–888.
• [BM] F. Bourgeois and K. Mohnke, Coherent orientations in symplectic field theory, Math. Z. 248 (2004), 123–146.
• [BN] F. Bourgeois and K. Niederkrüger, Towards a good definition of algebraically overtwisted, Expo. Math. 28 (2010), 85–100.
• [CL] K. Cieliebak and J. Latschev, “The role of string topology in symplectic field theory” in New Perspectives and Challenges in Symplectic Field Theory (Stanford, Calif., 2007), CRM Proc. Lecture Notes 49, Amer. Math. Soc., Providence, 2009, 113–146.
• [CGH1] V. Colin, P. Ghiggini, and K. Honda, Embedded contact homology and open book decompositions, preprint, arXiv:1008.2734v1 [math.SG].
• [CGH2] V. Colin, P. Ghiggini, and K. Honda, The equivalence of Heegaard Floer homology and embedded contact homology, III: From hat to plus, preprint, arXiv:1208.1526v1 [math.GT].
• [CGiH1] V. Colin, E. Giroux, and K. Honda, “On the coarse classification of tight contact structures” in Topology and Geometry of Manifolds (Athens, Ga., 2001), Proc. Sympos. Pure Math. 71, Amer. Math. Soc., Providence, 2003, 109–120.
• [CGiH2] V. Colin, E. Giroux, and K. Honda, Finitude homotopique et isotopique des structures de contact tendues, Publ. Math. Inst. Hautes Études Sci. 109 (2009), 245–293.
• [D] S. K. Donaldson, Connections, cohomology and the intersection forms of $4$-manifolds, J. Differential Geom. 24 (1986), 275–341.
• [Dr] D. L. Dragnev, Fredholm theory and transversality for noncompact pseudoholomorphic maps in symplectizations, Comm. Pure Appl. Math. 57 (2004), 726–763.
• [E1] Y. Eliashberg, Classification of overtwisted contact structures on $3$-manifolds, Invent. Math. 98 (1989), 623–637.
• [E2] 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.
• [E3] Y. Eliashberg, A few remarks about symplectic filling, Geom. Topol. 8 (2004), 277–293.
• [EGH] Y. Eliashberg, A. Givental, and H. Hofer, “Introduction to symplectic field theory” in GAFA 2000 (Tel Aviv, 1999), Geom. Funct. Anal. 2000, Special Volume, Part II, Birkhäuser, Basel, 2000, 560–673.
• [Et1] J. B. Etnyre, Planar open book decompositions and contact structures, Int. Math. Res. Not. 79 (2004), 4255–4267.
• [Et2] J. B. Etnyre, “Lectures on open book decompositions and contact structures” in Floer Homology, Gauge Theory, and Low-dimensional Topology (Budapest, 2004), Clay Math. Proc. 5, Amer. Math. Soc., Providence, 2006, 103–141.
• [EtH] J. B. Etnyre and K. Honda, On symplectic cobordisms, Math. Ann. 323 (2002), 31–39.
• [EVH] J. B. Etnyre and J. Van Horn-Morris, Fibered transverse knots and the Bennequin bound, Int. Math. Res. Not. IMRN 7 (2011), 1483–1509.
• [EVV] J. B. Etnyre and D. S. Vela-Vick, Torsion and open book decompositions, Int. Math. Res. Not. IMRN 22 (2010), 4385–4398.
• [Ga] D. T. Gay, Four-dimensional symplectic cobordisms containing three-handles, Geom. Topol. 10 (2006), 1749–1759.
• [Ge1] H. Geiges, Constructions of contact manifolds, Math. Proc. Cambridge Philos. Soc. 121 (1997), 455–464.
• [Ge2] H. Geiges, An Introduction to Contact Topology, Cambridge Stud. Adv. Math. 109, Cambridge Univ. Press, Cambridge, 2008.
• [GH] P. Ghiggini and K. Honda, Giroux torsion and twisted coefficients, preprint, arXiv:0804.1568v1 [math.GT].
• [GHVH] P. Ghiggini, K. Honda, and J. Van Horn-Morris, The vanishing of the contact invariant in the presence of torsion, preprint, arXiv:0706.1602v2 [math.GT].
• [Gi1] E. Giroux, Une structure de contact, même tendue, est plus ou moins tordue, Ann. Sci. Éc. Norm. Supér. (4) 27 (1994), 697–705.
• [Gi2] E. Giroux, Structures de contact sur les variétés fibrées en cercles audessus d’une surface, Comment. Math. Helv. 76 (2001), 218–262.
• [Gi3] E. Giroux, Links and contact structures, conference lecture at “Georgia International Topology Conference,” Athens, Georgia, 2001.
• [Gr1] M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307–347.
• [Gr2] M. Gromov, Partial Differential Relations, Ergeb. Math. Grenzgeb. (3) 9, Springer, Berlin, 1986.
• [H1] H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math. 114 (1993), 515–563.
• [H2] H. Hofer, “Holomorphic curves and real three-dimensional dynamics” in GAFA 2000 (Tel Aviv, 1999), Geom. Funct. Anal. 2000, Special Volume, Part II, Birkhäuser, Basel, 2000, 674–704.
• [H3] H. Hofer, “A general Fredholm theory and applications” in Current Developments in Mathematics, 2004, International Press, Somerville, Mass., 2006, 1–71.
• [HWZ1] H. Hofer, K. Wysocki, and E. Zehnder, A characterisation of the tight three-sphere, Duke Math. J. 81 (1995), 159–226.
• [HWZ2] 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.
• [HWZ3] H. Hofer, K. Wysocki, and E. Zehnder, Properties of pseudoholomorphic curves in symplectisations, I: Asymptotics, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), 337–379.
• [HWZ4] H. Hofer, K. Wysocki, and E. Zehnder, “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.
• [HWZ5] H. Hofer, K. Wysocki, and E. Zehnder, The dynamics on three-dimensional strictly convex energy surfaces, Ann. of Math. (2) 148 (1998), 197–289.
• [HWZ6] H. Hofer, K. Wysocki, and E. Zehnder, Finite energy foliations of tight three-spheres and Hamiltonian dynamics, Ann. of Math. (2) 157 (2003), 125–255.
• [HKM] K. Honda, W. H. Kazez, and G. Matic, Contact structures, sutured Floer homology and TQFT, preprint, arXiv:0807.2431v1 [math.GT].
• [Hu1] M. Hutchings, An index inequality for embedded pseudoholomorphic curves in symplectizations, J. Eur. Math. Soc. (JEMS) 4 (2002), 313–361.
• [Hu2] M. Hutchings, “Embedded contact homology and its applications” in Proceedings of the International Congress of Mathematicians, Vol. II (Hyderabad, 2010), Hindustan Book Agency, New Delhi, 2010, 1022–1041.
• [HuS] M. Hutchings and M. Sullivan, Rounding corners of polygons and the embedded contact homology of $T^{3}$, Geom. Topol. 10 (2006), 169–266.
• [HuT1] M. Hutchings and C. H. Taubes, Gluing pseudoholomorphic curves along branched covered cylinders. I, J. Symplectic Geom. 5 (2007), 43–137.
• [HuT2] M. Hutchings and C. H. Taubes, Gluing pseudoholomorphic curves along branched covered cylinders. II, J. Symplectic Geom. 7 (2009), 29–133.
• [HuT3] M. Hutchings and C. H. Taubes, Proof of the Arnold chord conjecture in three dimensions, II, preprint, arXiv:1111.3324v2 [math.SG].
• [KM] P. B. Kronheimer and T. S. Mrowka, Monopoles and contact structures, Invent. Math. 130 (1997), 209–255.
• [KLT1] Ç. Kutluhan, Y.-J. Lee, and C. H. Taubes, HF$\,=\,$HM I: Heegaard Floer homology and Seiberg-Witten Floer homology, preprint, arXiv:1007.1979v5 [math.GT].
• [KLT2] Ç. Kutluhan, Y.-J. Lee, and C. H. Taubes, HF$\,=\,$HM II: Reeb orbits and holomorphic curves for the ech/Heegard-Floer correspondence, preprint, arXiv:1008.1595v2 [math.GT].
• [KLT3] Ç. Kutluhan, Y.-J. Lee, and C. H. Taubes, HF$\,=\,$HM V: Seiberg-Witten-Floer homology and handle addition, preprint, arXiv:1204.0115v2 [math.GT].
• [LW] J. Latschev and C. Wendl, Algebraic torsion in contact manifolds, with an appendix by M. Hutchings, Geom. Funct. Anal. 21 (2011), 1144–1195.
• [L1] P. Lisca, Symplectic fillings and positive scalar curvature, Geom. Topol. 2 (1998), 103–116.
• [L2] P. Lisca, On symplectic fillings of $3$-manifolds, Turkish J. Math. 23 (1999), 151–159.
• [LVHW] S. Lisi, J. Van Horn-Morris, and C. Wendl, On symplectic fillings of spinal open book decompositions, in preparation.
• [Lu] R. Lutz, Structures de contact sur les fibrés principaux en cercles de dimension trois, Ann. Inst. Fourier (Grenoble) 27 (1977), 1–15.
• [M] P. Massot, Infinitely many universally tight torsion free contact structures with vanishing Ozsváth-Szabó contact invariants, Math. Ann. 353 (2012), 1351–1376.
• [MNW] P. Massot, K. Niederkrüger, and C. Wendl, Weak and strong fillability of higher dimensional contact manifolds, Invent. Math. 192 (2013), 287–373.
• [Ma] D. V. Mathews, Sutured Floer homology, sutured TQFT and noncommutative QFT, Algebr. Geom. Topol. 11 (2011), 2681–2739.
• [Mc1] D. McDuff, The structure of rational and ruled symplectic $4$-manifolds, J. Amer. Math. Soc. 3 (1990), 679–712.
• [Mc2] D. McDuff, Symplectic manifolds with contact type boundaries, Invent. Math. 103 (1991), 651–671.
• [Mn] A. Momin, Intersection theory in $3$-dimensional linearized contact homology, Ph.D. dissertation, New York University, New York, N.Y., 2008.
• [Mo] E. Mora, Pseudoholomorphic cylinders in symplectisations, Ph.D. dissertation, New York University, New York, N.Y., 2003.
• [Mi] A. Mori, Reeb foliations on $S^{5}$ and contact $5$-manifolds violating the Thurston-Bennequin inequality, preprint, arXiv:0906.3237v2 [math.GT].
• [N] K. Niederkrüger, The plastikstufe—a generalization of the overtwisted disk to higher dimensions, Algebr. Geom. Topol. 6 (2006), 2473–2508.
• [NW] K. Niederkrüger and C. Wendl, Weak symplectic fillings and holomorphic curves, Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), 801–853.
• [OSS] P. Ozsváth, A. Stipsicz, and Z. Szabó, Planar open books and Floer homology, Int. Math. Res. Not. 54 (2005), 3385–3401.
• [P] O. Plamenevskaya, On Legendrian surgeries between lens spaces, J. Symplectic Geom. 10 (2012), 165–181.
• [PVH] O. Plamenevskaya and J. Van Horn-Morris, Planar open books, monodromy factorizations and symplectic fillings, Geom. Topol. 14 (2010), 2077–2101.
• [Sc] M. Schwarz, Cohomology operations from $S^{1}$-cobordisms in Floer homology, Ph.D. Dissertation, Eidgenössische Technische Hochschule Zürich, Zürich, Switzerland, 1995.
• [Si1] R. Siefring, Relative asymptotic behavior of pseudoholomorphic half-cylinders, Comm. Pure Appl. Math. 61 (2008), 1631–1684.
• [Si2] R. Siefring, Intersection theory of punctured pseudoholomorphic curves, Geom. Topol. 15 (2011), 2351–2457.
• [T1] C. H. Taubes, The structure of pseudo-holomorphic subvarieties for a degenerate almost complex structure and symplectic form on $S^{1}\times B^{3}$, Geom. Topol. 2 (1998), 221–332.
• [T2] C. H. Taubes, Embedded contact homology and Seiberg-Witten Floer cohomology, I, Geom. Topol. 14 (2010), 2497–2581.
• [T3] C. H. Taubes, Embedded contact homology and Seiberg-Witten Floer cohomology, V, Geom. Topol. 14 (2010), 2961–3000.
• [VB] J. von Bergmann, Embedded H-holomorphic maps and open book decompositions, preprint, arXiv:0907.3939v1 [math.SG].
• [W1] C. Wendl, Finite energy foliations on overtwisted contact manifolds, Geom. Topol. 12 (2008), 531–616.
• [W2] C. Wendl, Automatic transversality and orbifolds of punctured holomorphic curves in dimension four, Comment. Math. Helv. 85 (2010), 347–407.
• [W3] C. Wendl, Open book decompositions and stable Hamiltonian structures, Expo. Math. 28 (2010), 187–199.
• [W4] C. Wendl, Strongly fillable contact manifolds and $J$-holomorphic foliations, Duke Math. J. 151 (2010), 337–384.
• [W5] C. Wendl, Non-exact symplectic cobordisms between contact $3$-manifolds, preprint, arXiv:1008.2456v3 [math.SG].
• [W6] C. Wendl, Finite energy foliations and surgery on transverse links, Ph.D. dissertation, New York University, New York, N.Y., 2005.
• [Y] M.-L. Yau, Vanishing of the contact homology of overtwisted contact $3$-manifolds, with an appendix by Y. Eliashberg, Bull. Inst. Math. Acad. Sin. (N.S.) 1 (2006), 211–229.