Geometric criteria for overtwistedness

Casals, Roger; Murphy, Emmy; Presas, Francisco

doi:10.1090/jams/917

Geometric criteria for overtwistedness

Authors: Roger Casals, Emmy Murphy and Francisco Presas
Journal: J. Amer. Math. Soc. 32 (2019), 563-604
MSC (2010): Primary 57R17; Secondary 53D10, 53D15
DOI: https://doi.org/10.1090/jams/917
Published electronically: January 3, 2019
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this article we establish efficient geometric criteria to decide whether a contact manifold is overtwisted. Starting with the original definition, we first relate overtwisted disks in different dimensions and show that a manifold is overtwisted if and only if the Legendrian unknot admits a loose chart. Then we characterize overtwistedness in terms of the monodromy of open book decompositions and contact surgeries. Finally, we provide several applications of these geometric criteria.

[1] V. I. Arnol′d and A. B. Givental′, Symplectic geometry [ MR0842908 (88b:58044)], Dynamical systems, IV, Encyclopaedia Math. Sci., vol. 4, Springer, Berlin, 2001, pp. 1–138. MR 1866631
[2] R. Avdek, Liouville hypersurfaces and connect sum cobordisms, arXiv:1204.3145.
[3] Jonathan Bowden, Diarmuid Crowley, and András I. Stipsicz, Contact structures on 𝑀×𝑆², Math. Ann. 358 (2014), no. 1-2, 351–359. MR 3158000, https://doi.org/10.1007/s00208-013-0963-9
[4] Matthew Strom Borman, Yakov Eliashberg, and Emmy Murphy, Existence and classification of overtwisted contact structures in all dimensions, Acta Math. 215 (2015), no. 2, 281–361. MR 3455235, https://doi.org/10.1007/s11511-016-0134-4
[5] Frédéric Bourgeois, Odd dimensional tori are contact manifolds, Int. Math. Res. Not. 30 (2002), 1571–1574. MR 1912277, https://doi.org/10.1155/S1073792802205048
[6] Frédéric Bourgeois and Klaus Niederkrüger, Towards a good definition of algebraically overtwisted, Expo. Math. 28 (2010), no. 1, 85–100. MR 2606237, https://doi.org/10.1016/j.exmath.2009.06.001
[7] Frédéric Bourgeois and Otto van Koert, Contact homology of left-handed stabilizations and plumbing of open books, Commun. Contemp. Math. 12 (2010), no. 2, 223–263. MR 2646902, https://doi.org/10.1142/S0219199710003762
[8] R. Casals, Overtwisted disks and exotic symplectic structures, arXiv:1402.7099.
[9] R. Casals, Contact fibrations over the -disk, PhD Thesis, Univ. Autónoma de Madrid, 2015.
[10] Roger Casals and Emmy Murphy, Contact topology from the loose viewpoint, Proceedings of the Gökova Geometry-Topology Conference 2015, Gökova Geometry/Topology Conference (GGT), Gökova, 2016, pp. 81–115. MR 3526839
[11] R. Casals and E. Murphy, Legendrian fronts for affine varieties, Duke Math. J., to appear.
[12] Roger Casals and Francisco Presas, A remark on the Reeb flow for spheres, J. Symplectic Geom. 12 (2014), no. 4, 657–671. MR 3333025
[13] Roger Casals and Francisco Presas, On the strong orderability of overtwisted 3-folds, Comment. Math. Helv. 91 (2016), no. 2, 305–316. MR 3493373, https://doi.org/10.4171/CMH/387
[14] Roger Casals, Dishant M. Pancholi, and Francisco Presas, Almost contact 5-manifolds are contact, Ann. of Math. (2) 182 (2015), no. 2, 429–490. MR 3418523, https://doi.org/10.4007/annals.2015.182.2.2
[15] Roger Casals, Francisco Presas, and Sheila Sandon, Small positive loops on overtwisted manifolds, J. Symplectic Geom. 14 (2016), no. 4, 1013–1031. MR 3601882, https://doi.org/10.4310/JSG.2016.v14.n4.a2
[16] Kai Cieliebak and Yakov Eliashberg, From Stein to Weinstein and back, American Mathematical Society Colloquium Publications, vol. 59, American Mathematical Society, Providence, RI, 2012. Symplectic geometry of affine complex manifolds. MR 3012475
[17] V. Colin, Livres ouverts en géométrie de contact, Astérisque, Sém. Bourbaki 59 (2006), 91-118.
[18] Georgios Dimitroglou Rizell and Jonathan David Evans, Exotic spheres and the topology of symplectomorphism groups, J. Topol. 8 (2015), no. 2, 586–602. MR 3356772, https://doi.org/10.1112/jtopol/jtv007
[19] Fan Ding, Hansjörg Geiges, and András I. Stipsicz, Surgery diagrams for contact 3-manifolds, Turkish J. Math. 28 (2004), no. 1, 41–74. MR 2056760
[20] Yakov Eliashberg, Sang Seon Kim, and Leonid Polterovich, Geometry of contact transformations and domains: orderability versus squeezing, Geom. Topol. 10 (2006), 1635–1747. MR 2284048, https://doi.org/10.2140/gt.2006.10.1635
[21] Y. Eliashberg, A. Givental, and H. Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. Special Volume (2000), 560–673. GAFA 2000 (Tel Aviv, 1999). MR 1826267, https://doi.org/10.1007/978-3-0346-0425-3_4
[22] Y. Eliashberg, Classification of overtwisted contact structures on 3-manifolds, Invent. Math. 98 (1989), no. 3, 623–637. MR 1022310, https://doi.org/10.1007/BF01393840
[23] Yakov Eliashberg, Topological characterization of Stein manifolds of dimension >2, Internat. J. Math. 1 (1990), no. 1, 29–46. MR 1044658, https://doi.org/10.1142/S0129167X90000034
[24] Yakov Eliashberg, Classification of contact structures on 𝐑³, Internat. Math. Res. Notices 3 (1993), 87–91. MR 1208828, https://doi.org/10.1155/S107379289300008X
[25] Yakov Eliashberg, Recent advances in symplectic flexibility, Bull. Amer. Math. Soc. (N.S.) 52 (2015), no. 1, 1–26. MR 3286479, https://doi.org/10.1090/S0273-0979-2014-01470-3
[26] Y. Eliashberg and N. Mishachev, Introduction to the ℎ-principle, Graduate Studies in Mathematics, vol. 48, American Mathematical Society, Providence, RI, 2002. MR 1909245
[27] Y. Eliashberg and E. Murphy, Making cobordisms symplectic, arXiv:1504.06312.
[28] Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, Lagrangian intersection Floer theory: anomaly and obstruction. Part I, AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009. MR 2553465
[29] Hansjörg Geiges, An introduction to contact topology, Cambridge Studies in Advanced Mathematics, vol. 109, Cambridge University Press, Cambridge, 2008. MR 2397738
[30] Hansjörg Geiges, Constructions of contact manifolds, Math. Proc. Cambridge Philos. Soc. 121 (1997), no. 3, 455–464. MR 1434654, https://doi.org/10.1017/S0305004196001260
[31] Emmanuel Giroux, Géométrie de contact: de la dimension trois vers les dimensions supérieures, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 405–414 (French, with French summary). MR 1957051
[32] Emmanuel Giroux, Sur la géométrie et la dynamique des transformations de contact (d’après Y. Eliashberg, L. Polterovich et al.), Astérisque 332 (2010), Exp. No. 1004, viii, 183–220 (French, with French summary). Séminaire Bourbaki. Volume 2008/2009. Exposés 997–1011. MR 2648679
[33] Emmanuel Giroux and John Pardon, Existence of Lefschetz fibrations on Stein and Weinstein domains, Geom. Topol. 21 (2017), no. 2, 963–997. MR 3626595, https://doi.org/10.2140/gt.2017.21.963
[34] M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307–347. MR 809718, https://doi.org/10.1007/BF01388806
[35] Mikhael Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 9, Springer-Verlag, Berlin, 1986. MR 864505
[36] Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
[37] Ko Honda, William H. Kazez, and Gordana Matić, Right-veering diffeomorphisms of compact surfaces with boundary, Invent. Math. 169 (2007), no. 2, 427–449. MR 2318562, https://doi.org/10.1007/s00222-007-0051-4
[38] Michael Hutchings, Quantitative embedded contact homology, J. Differential Geom. 88 (2011), no. 2, 231–266. MR 2838266
[39] Yang Huang, On plastikstufe, bordered Legendrian open book and overtwisted contact structures, J. Topol. 10 (2017), no. 3, 720–743. MR 3665409, https://doi.org/10.1112/topo.12020
[40] Otto van Koert, Lecture notes on stabilization of contact open books, Münster J. Math. 10 (2017), no. 2, 425–455. MR 3725503
[41] Peter Kronheimer and Tomasz Mrowka, Monopoles and three-manifolds, New Mathematical Monographs, vol. 10, Cambridge University Press, Cambridge, 2007. MR 2388043
[42] Patrick Massot, Klaus Niederkrüger, and Chris Wendl, Weak and strong fillability of higher dimensional contact manifolds, Invent. Math. 192 (2013), no. 2, 287–373. MR 3044125, https://doi.org/10.1007/s00222-012-0412-5
[43] Tomasz Mrowka and Yann Rollin, Legendrian knots and monopoles, Algebr. Geom. Topol. 6 (2006), 1–69. MR 2199446, https://doi.org/10.2140/agt.2006.6.1
[44] E. Murphy, Loose Legendrian embeddings in high dimensional contact manifolds, arXiv:1201.2245.
[45] Emmy Murphy, Klaus Niederkrüger, Olga Plamenevskaya, and András I. Stipsicz, Loose Legendrians and the plastikstufe, Geom. Topol. 17 (2013), no. 3, 1791–1814. MR 3073936, https://doi.org/10.2140/gt.2013.17.1791
[46] L. Polterovich, The surgery of Lagrange submanifolds, Geom. Funct. Anal. 1 (1991), no. 2, 198–210. MR 1097259, https://doi.org/10.1007/BF01896378
[47] Klaus Niederkrüger, The plastikstufe—a generalization of the overtwisted disk to higher dimensions, Algebr. Geom. Topol. 6 (2006), 2473–2508. MR 2286033, https://doi.org/10.2140/agt.2006.6.2473
[48] Klaus Niederkrüger and Francisco Presas, Some remarks on the size of tubular neighborhoods in contact topology and fillability, Geom. Topol. 14 (2010), no. 2, 719–754. MR 2602849, https://doi.org/10.2140/gt.2010.14.719
[49] Klaus Niederkrüger and Otto van Koert, Every contact manifolds can be given a nonfillable contact structure, Int. Math. Res. Not. IMRN 23 (2007), Art. ID rnm115, 22. MR 2380008, https://doi.org/10.1093/imrn/rnm115
[50] Burak Ozbagci and András I. Stipsicz, Surgery on contact 3-manifolds and Stein surfaces, Bolyai Society Mathematical Studies, vol. 13, Springer-Verlag, Berlin; János Bolyai Mathematical Society, Budapest, 2004. MR 2114165
[51] P. Seidel, Floer homology and the symplectic isotopy problem, PhD Thesis, Oxford University, 1997.
[52] Paul Seidel, Lagrangian two-spheres can be symplectically knotted, J. Differential Geom. 52 (1999), no. 1, 145–171. MR 1743463
[53] Paul Seidel, Fukaya categories and Picard-Lefschetz theory, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008. MR 2441780
[54] Andy Wand, Tightness is preserved by Legendrian surgery, Ann. of Math. (2) 182 (2015), no. 2, 723–738. MR 3418529, https://doi.org/10.4007/annals.2015.182.2.8
[55] Alan Weinstein, Contact surgery and symplectic handlebodies, Hokkaido Math. J. 20 (1991), no. 2, 241–251. MR 1114405, https://doi.org/10.14492/hokmj/1381413841