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
MathSciNet review:
3904160
Full-text PDF
View in AMS MathViewer
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.
- 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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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
- 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 https://doi.org/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 https://doi.org/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 https://doi.org/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
- 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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/10.1142/S0129167X90000034
- Yakov Eliashberg, Classification of contact structures on $\mathbf R^3$, Internat. Math. Res. Notices 3 (1993), 87–91. MR 1208828, DOI https://doi.org/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 https://doi.org/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
- 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
- Hansjörg Geiges, An introduction to contact topology, Cambridge Studies in Advanced Mathematics, vol. 109, Cambridge University Press, Cambridge, 2008. MR 2397738
- Hansjörg Geiges, Constructions of contact manifolds, Math. Proc. Cambridge Philos. Soc. 121 (1997), no. 3, 455–464. MR 1434654, DOI https://doi.org/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 https://doi.org/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 https://doi.org/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
- 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 https://doi.org/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 https://doi.org/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 https://doi.org/10.17879/70299609615
- Peter Kronheimer and Tomasz Mrowka, Monopoles and three-manifolds, New Mathematical Monographs, vol. 10, Cambridge University Press, Cambridge, 2007. MR 2388043
- 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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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
- 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
- Andy Wand, Tightness is preserved by Legendrian surgery, Ann. of Math. (2) 182 (2015), no. 2, 723–738. MR 3418529, DOI https://doi.org/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 https://doi.org/10.14492/hokmj/1381413841
Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 57R17, 53D10, 53D15
Retrieve articles in all journals with MSC (2010): 57R17, 53D10, 53D15
Additional Information
Roger Casals
Affiliation:
Department of Mathematics, University of California Davis, Shields Avenue, Davis, California 95616
MR Author ID:
1096004
Email:
casals@math.ucdavis.edu
Emmy Murphy
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, Illinois 60208
Email:
e_murphy@math.northwestern.edu
Francisco Presas
Affiliation:
Instituto de Ciencias Matemáticas CSIC, C. Nicolás Cabrera, 13 28049 Madrid, Spain
MR Author ID:
678330
Email:
fpresas@icmat.es
Received by editor(s):
January 23, 2017
Received by editor(s) in revised form:
September 12, 2018, and October 8, 2018
Published electronically:
January 3, 2019
Additional Notes:
The first author was supported by NSF grant DMS-1841913 and a BBVA Research Fellowship.
The second author was supported by NSF grant DMS-1510305 and a Sloan Research Fellowship.
The third author was supported by Spanish Research Projects SEV–2015–0554, MTM2016–79400–P, and MTM2015–72876–EXP
Article copyright:
© Copyright 2019
American Mathematical Society