Geometric criteria for overtwistedness

Casals, Roger; Murphy, Emmy; Presas, Francisco

doi:10.1090/jams/917

Geometric criteria for overtwistedness
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by Roger Casals, Emmy Murphy and Francisco Presas;

J. Amer. Math. Soc. 32 (2019), 563-604

DOI: https://doi.org/10.1090/jams/917

Published electronically: January 3, 2019

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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.

References

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
R. Avdek, Liouville hypersurfaces and connect sum cobordisms, arXiv:1204.3145.
Jonathan Bowden, Diarmuid Crowley, and András I. Stipsicz, Contact structures on $M\times S^2$, Math. Ann. 358 (2014), no. 1-2, 351–359. MR 3158000, DOI 10.1007/s00208-013-0963-9
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, DOI 10.1007/s11511-016-0134-4
Frédéric Bourgeois, Odd dimensional tori are contact manifolds, Int. Math. Res. Not. 30 (2002), 1571–1574. MR 1912277, DOI 10.1155/S1073792802205048
Frédéric Bourgeois and Klaus Niederkrüger, Towards a good definition of algebraically overtwisted, Expo. Math. 28 (2010), no. 1, 85–100. MR 2606237, DOI 10.1016/j.exmath.2009.06.001
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, DOI 10.1142/S0219199710003762
R. Casals, Overtwisted disks and exotic symplectic structures, arXiv:1402.7099.
R. Casals, Contact fibrations over the $2$–disk, PhD Thesis, Univ. Autónoma de Madrid, 2015.
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
R. Casals and E. Murphy, Legendrian fronts for affine varieties, Duke Math. J., to appear.
Roger Casals and Francisco Presas, A remark on the Reeb flow for spheres, J. Symplectic Geom. 12 (2014), no. 4, 657–671. MR 3333025, DOI 10.4310/JSG.2014.v12.n4.a1
Roger Casals and Francisco Presas, On the strong orderability of overtwisted 3-folds, Comment. Math. Helv. 91 (2016), no. 2, 305–316. MR 3493373, DOI 10.4171/CMH/387
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, DOI 10.4007/annals.2015.182.2.2
Roger Casals, Francisco Presas, and Sheila Sandon, Small positive loops on overtwisted manifolds, J. Symplectic Geom. 14 (2016), no. 4, 1013–1031. MR 3601882, DOI 10.4310/JSG.2016.v14.n4.a2
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, DOI 10.1090/coll/059
V. Colin, Livres ouverts en géométrie de contact, Astérisque, Sém. Bourbaki 59 (2006), 91–118.
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, DOI 10.1112/jtopol/jtv007
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
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, DOI 10.2140/gt.2006.10.1635
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, DOI 10.1007/978-3-0346-0425-3_{4}
Y. Eliashberg, Classification of overtwisted contact structures on $3$-manifolds, Invent. Math. 98 (1989), no. 3, 623–637. MR 1022310, DOI 10.1007/BF01393840
Yakov Eliashberg, Topological characterization of Stein manifolds of dimension $>2$, Internat. J. Math. 1 (1990), no. 1, 29–46. MR 1044658, DOI 10.1142/S0129167X90000034
Yakov Eliashberg, Classification of contact structures on $\mathbf R^3$, Internat. Math. Res. Notices 3 (1993), 87–91. MR 1208828, DOI 10.1155/S107379289300008X
Yakov Eliashberg, Recent advances in symplectic flexibility, Bull. Amer. Math. Soc. (N.S.) 52 (2015), no. 1, 1–26. MR 3286479, DOI 10.1090/S0273-0979-2014-01470-3
Y. Eliashberg and N. Mishachev, Introduction to the $h$-principle, Graduate Studies in Mathematics, vol. 48, American Mathematical Society, Providence, RI, 2002. MR 1909245, DOI 10.1090/gsm/048
Y. Eliashberg and E. Murphy, Making cobordisms symplectic, arXiv:1504.06312.
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, DOI 10.1090/amsip/046.1
Hansjörg Geiges, An introduction to contact topology, Cambridge Studies in Advanced Mathematics, vol. 109, Cambridge University Press, Cambridge, 2008. MR 2397738, DOI 10.1017/CBO9780511611438
Hansjörg Geiges, Constructions of contact manifolds, Math. Proc. Cambridge Philos. Soc. 121 (1997), no. 3, 455–464. MR 1434654, DOI 10.1017/S0305004196001260
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
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
Emmanuel Giroux and John Pardon, Existence of Lefschetz fibrations on Stein and Weinstein domains, Geom. Topol. 21 (2017), no. 2, 963–997. MR 3626595, DOI 10.2140/gt.2017.21.963
M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307–347. MR 809718, DOI 10.1007/BF01388806
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, DOI 10.1007/978-3-662-02267-2
Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
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, DOI 10.1007/s00222-007-0051-4
Michael Hutchings, Quantitative embedded contact homology, J. Differential Geom. 88 (2011), no. 2, 231–266. MR 2838266
Yang Huang, On plastikstufe, bordered Legendrian open book and overtwisted contact structures, J. Topol. 10 (2017), no. 3, 720–743. MR 3665409, DOI 10.1112/topo.12020
Otto van Koert, Lecture notes on stabilization of contact open books, Münster J. Math. 10 (2017), no. 2, 425–455. MR 3725503, DOI 10.17879/70299609615
Peter Kronheimer and Tomasz Mrowka, Monopoles and three-manifolds, New Mathematical Monographs, vol. 10, Cambridge University Press, Cambridge, 2007. MR 2388043, DOI 10.1017/CBO9780511543111
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, DOI 10.1007/s00222-012-0412-5
Tomasz Mrowka and Yann Rollin, Legendrian knots and monopoles, Algebr. Geom. Topol. 6 (2006), 1–69. MR 2199446, DOI 10.2140/agt.2006.6.1
E. Murphy, Loose Legendrian embeddings in high dimensional contact manifolds, arXiv:1201.2245.
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, DOI 10.2140/gt.2013.17.1791
L. Polterovich, The surgery of Lagrange submanifolds, Geom. Funct. Anal. 1 (1991), no. 2, 198–210. MR 1097259, DOI 10.1007/BF01896378
Klaus Niederkrüger, The plastikstufe—a generalization of the overtwisted disk to higher dimensions, Algebr. Geom. Topol. 6 (2006), 2473–2508. MR 2286033, DOI 10.2140/agt.2006.6.2473
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, DOI 10.2140/gt.2010.14.719
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, DOI 10.1093/imrn/rnm115
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, DOI 10.1007/978-3-662-10167-4
P. Seidel, Floer homology and the symplectic isotopy problem, PhD Thesis, Oxford University, 1997.
Paul Seidel, Lagrangian two-spheres can be symplectically knotted, J. Differential Geom. 52 (1999), no. 1, 145–171. MR 1743463
Paul Seidel, Fukaya categories and Picard-Lefschetz theory, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008. MR 2441780, DOI 10.4171/063
Andy Wand, Tightness is preserved by Legendrian surgery, Ann. of Math. (2) 182 (2015), no. 2, 723–738. MR 3418529, DOI 10.4007/annals.2015.182.2.8
Alan Weinstein, Contact surgery and symplectic handlebodies, Hokkaido Math. J. 20 (1991), no. 2, 241–251. MR 1114405, DOI 10.14492/hokmj/1381413841