Abstract
In this article we show that in any dimension there exist infinitely many pairs of formally contact isotopic isocontact embeddings into the standard contact sphere which are not contact isotopic. This is the first example of rigidity for contact submanifolds in higher dimensions. The contact embeddings are constructed via contact push-offs of higher-dimensional Legendrian submanifolds, a construction that generalizes the union of the positive and negative transverse push-offs of a Legendrian knot to higher dimensions.
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References
V.I. Arnol’ d. Singularities of Caustics and Wave Fronts, volume 62 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht (1990).
V.I. Arnol’ d. Some remarks on symplectic monodromy of Milnor fibrations. In: The Floer Memorial Volume, volume 133 of Progress in Mathematics, Birkhäuser, Basel (1995), pp. 99–103.
V.I. Arnol’ d and A.B. Givental’. Symplectic geometry [MR0842908 (88b:58044)]. In: Dynamical Systems, IV, volume 4 of Encyclopaedia Mathematical Sciences, Springer, Berlin (2001), pp. 1–138.
D. Auroux and I. Smith. Lefschetz pencils, branched covers and symplectic invariants. In: Symplectic 4-Manifolds and Algebraic Surfaces, volume 1938 of Lecture Notes in Mathematics, Springer, Berlin (2008), pp. 1–53.
R. Avdek. Liouville hypersurfaces and connect sum cobordisms. ArXiv e-prints (April 2012).
D. Bennequin. Entrelacements et équations de Pfaff. In: Third Schnepfenried Geometry Conference, Vol. 1 (Schnepfenried, 1982), volume 107 of Astérisque, Society of Mathematics France, Paris (1983), pp. 87–161.
J.S. Birman and W.W. Menasco. Stabilization in the braid groups. II. Transversal simplicity of knots. Geom. Topol., 10 (2006), 1425–1452 (electronic)
M. Borman, Y. Eliashberg, and E. Murphy. Existence and classification of overtwisted contact structures in all dimensions. ArXiv e-prints (April 2014)
M.S. Borman, Y. Eliashberg, and E. Murphy. Existence and classification of overtwisted contact structures in all dimensions. Acta Math., (2)215 (2015), 281–361
F. Bourgeois, T. Ekholm, and Y. Eliashberg. Effect of Legendrian surgery. Geom. Topol., (1)16 (2012), 301–389. With an appendix by Sheel Ganatra and Maksim Maydanskiy
F. Bourgeois and A. Oancea. An exact sequence for contact- and symplectic homology. Invent. Math., (3)175 (2009), 611–680
F. Bourgeois, J.M. Sabloff, and L. Traynor. Lagrangian cobordisms via generating families: construction and geography. Algebr. Geom. Topol., (4)15 (2015), 2439–2477
R. Casals and E. Murphy. Legendrian fronts for affine varieties. Duke Math. J., (2)168 (2019), 1–136
R. Casals, E. Murphy, and F. Presas. Geometric criteria for overtwistedness. J. Am. Math. Soc., (2)32 (2019), 563–604
K. Cieliebak and Y. Eliashberg. From Stein to Weinstein and Back, volume 59 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI (2012), Symplectic geometry of affine complex manifolds.
K. Cieliebak and Y. Eliashberg. Stein structures: existence and flexibility. In: Contact and Symplectic Topology, volume 26 of Bolyai Society Mathematical Studies, János Bolyai Math. Soc., Budapest (2014), pp. 357–388.
V. Colin. Livres ouverts en géométrie de contact (d’après Emmanuel Giroux). Astérisque, (317):Exp. No. 969, vii, 91–117, (2008), Séminaire Bourbaki. Vol. 2006/2007.
S.K. Donaldson. Symplectic submanifolds and almost-complex geometry. J. Differ. Geom., (4)44 (1996), 666–705
T. Ekholm, J. Etnyre, L. Ng, and M. Sullivan. Filtrations on the knot contact homology of transverse knots. Math. Ann., (4)355 (2013), 1561–1591
T. Ekholm, J. Etnyre, and M. Sullivan. The contact homology of Legendrian submanifolds in \({\mathbb{R}}^{2n+1}\). J. Differ. Geom., (2)71 (2005), 177–305
T. Ekholm, J. Etnyre, and M. Sullivan. Non-isotopic Legendrian submanifolds in \({\mathbb{R}}^{2n+1}\). J. Differ. Geom., (1)71 (2005), 85–128
Y. Eliashberg. Classification of overtwisted contact structures on \(3\)-manifolds. Invent. Math., (3)98 (1989), 623–637
Y. Eliashberg. Weinstein manifolds revisited. Modern Geometry: A Celebration of the Work of Simon Donaldson, Proceedings of Symposia in Pure Mathematics, (2018), 99
Y. Eliashberg, A. Givental, and H. Hofer. Introduction to symplectic field theory. Geometric and Functional Analysis, (Special Volume, Part II), (2000), GAFA 2000 (Tel Aviv, 1999), 560–673
Y. Eliashberg and N. Mishachev. Introduction to the \(h\)-Principle, volume 48 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2002)
Y. Eliashberg. Topological characterization of Stein manifolds of dimension > 2. Internat. J. Math., (1)1 (1990), 29–46
Y. Eliashberg. Recent advances in symplectic flexibility. Bull. Am. Math. Soc. (N.S.), (1)52 (2015), 1–26
Y. Eliashberg and M. Gromov. Convex symplectic manifolds. In. Several Complex Variables and Complex Geometry, Part 2 (Santa Cruz, CA, 1989), volume 52 of Proceedings of Symposia in Pure Mathematics, American Mathematical Society, Providence, RI (1991), pp. 135–162.
Y. Eliashberg and E. Murphy. Lagrangian caps. Geom. Funct. Anal., (5)23 (2013), 1483–1514
J. Epstein, D. Fuchs, and M. Meyer. Chekanov–Eliashberg invariants and transverse approximations of Legendrian knots. Pac. J. Math., (1)201 (2001), 89–106
J.B. Etnyre. Planar open book decompositions and contact structures. Int. Math. Res. Not., (79) (2004), 4255–4267
J.B. Etnyre. Legendrian and transversal knots. In. Handbook of Knot Theory. Elsevier B. V., Amsterdam (2005), pp. 105–185.
J.B. Etnyre and R. Furukawa. Braided embeddings of contact 3-manifolds in the standard contact 5-sphere. J. Topol., (2)10 (2017), 412–446
J.B. Etnyre and K. Honda. Cabling and transverse simplicity. Ann. Math. (2), (3)162 (2005), 1305–1333
J.B. Etnyre, D.J. LaFountain, and B. Tosun. Legendrian and transverse cables of positive torus knots. Geom. Topol., (3)16 (2012), 1639–1689
J.B. Etnyre and Y. Lekili. Embedding all contact 3-manifolds in a fixed contact 5-manifold. J. Lond. Math. Soc., (0)0
J.B. Etnyre, L.L. Ng, and V. Vértesi. Legendrian and transverse twist knots. J. Eur. Math. Soc. (JEMS), (3)15 (2013), 969–995
J.B. Etnyre and J. VanHorn-Morris. Fibered transverse knots and the Bennequin bound. Int. Math. Res. Not. IMRN, (7) (2011), 1483–1509
H. Federer and W.H. Fleming. Normal and integral currents. Ann. Math. (2), 72 (1960), 458–520
U. Frauenfelder, F. Schlenk, and O. vanKoert. Displaceability and the mean Euler characteristic. Kyoto J. Math., (4)52 (2012), 797–815
K. Fukaya. Mirror symmetry of abelian varieties and multi-theta functions. J. Algebraic Geom., (3)11 (2002), 393–512
H. Geiges. Constructions of contact manifolds. Math. Proc. Camb. Philos. Soc., (3)121 (1997), 455–464
H. Geiges. An Introduction to Contact Topology, volume 109 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2008).
E. Giroux. Convexité en topologie de contact. Comment. Math. Helv., (4)66 (1991), 637–677
E. Giroux. Géométrie de contact: de la dimension trois vers les dimensions supérieures. In Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing (2002), pp. 405–414.
J. Gonzalo. Branched covers and contact structures. Proc. Am. Math. Soc., (2)101 (1987), 347–352
M. Gromov. Pseudoholomorphic curves in symplectic manifolds. Invent. Math., (2)82 (1985), 307–347
M. Gromov. Partial Differential Relations, volume 9 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin (1986)
S. Harvey, K. Kawamuro, and O. Plamenevskaya. On transverse knots and branched covers. Int. Math. Res. Not. IMRN, (3) (2009), 512–546
K. Honda and Y. Huang. Bypass attachments in higher-dimensional contact topology. ArXiv e-prints (2018).
A. Ibort, D. Martínez-Torres, and F. Presas. On the construction of contact submanifolds with prescribed topology. J. Differ. Geom., (2)56 (2000), 235–283
M. Kwon and O. vanKoert. Brieskorn manifolds in contact topology. Bull. Lond. Math. Soc., (2)48 (2016), 173–241
O. Lazarev. Maximal contact and symplectic structures. ArXiv e-prints (2018).
R. Lutz. Sur quelques propriétés des formes differentielles en dimension trois. Thèse, Strasbourg (1971).
J. Martinet. Formes de contact sur les variétés de dimension \(3\). In. Proceedings of Liverpool Singularities Symposium, II (1969/1970), Lecture Notes in Math., Vol. 209. Springer, Berlin (1971), pp. 142–163.
J. Moser. On the volume elements on a manifold. Trans. Am. Math. Soc., 120 (1965), 286–294.
E. Murphy. Loose Legendrian Embeddings in High Dimensional Contact Manifolds. ArXiv e-prints (January 2012)
E. Murphy and K. Siegel. Subflexible symplectic manifolds. Geom. Topol., (4)22 (2018), 2367–2401
D. Nadler. Arboreal singularities. Geom. Topol., (2)21 (2017), 1231–1274
L. Ng. Combinatorial knot contact homology and transverse knots. Adv. Math., (6)227 (2011), 2189–2219
K. Niederkrüger and F. Presas. Some remarks on the size of tubular neighborhoods in contact topology and fillability. Geom. Topol., (2)14 (2010), 719–754
K. Niederkrüger and O. vanKoert. Every contact manifolds can be given a nonfillable contact structure. Int. Math. Res. Not. IMRN, (23)Art. ID rnm115 (2007), 22
P. Ozsváth, Z. Szabó, and D. Thurston. Legendrian knots, transverse knots and combinatorial Floer homology. Geom. Topol., (2)12 (2008), 941–980
F. Öztürk and K. Niederkrüger. Brieskorn manifolds as contact branched covers of spheres. Period. Math. Hungar., (1)54 (2007), 85–97
P.S. Pancholi. D.M. Iso-contact embeddings of manifolds in co-dimension 2. ArXiv e-prints (2018)
O. Plamenevskaya. Transverse knots and Khovanov homology. Math. Res. Lett., (4)13 (2006), 571–586
O. Plamenevskaya. Transverse knots, branched double covers and Heegaard Floer contact invariants. J. Symplectic Geom., (2)4 (2006), 149–170
F. Presas. A class of non-fillable contact structures. Geom. Topol., 11 (2007), 2203–2225
R.C. Randell. The homology of generalized Brieskorn manifolds. Topology, (4)14 (1975), 347–355
D. Rolfsen. Knots and links. Publish or Perish Inc., Berkeley, Calif. (1976), Mathematics Lecture Series, No. 7.
P. Seidel. Fukaya categories and Picard-Lefschetz theory. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2008)
L. Starkston. Arboreal singularities in weinstein skeleta. Selecta Math. to appear.
P. Uebele. Symplectic homology of some Brieskorn manifolds. Math. Z., (1-2)283 (2016), 243–274
O. vanKoert. Lecture notes on stabilization of contact open books. Münster J. Math., (2)10 (2017), 425–455
A. Weinstein. Contact surgery and symplectic handlebodies. Hokkaido Math. J., (2)20 (1991), 241–251
B. White. A new proof of the compactness theorem for integral currents. Comment. Math. Helv., (2)64 (1989), 207–220
Z. Zhou. Vanishing of Symplectic Homology and Obstruction to Flexible Fillability. ArXiv e-prints (October 2017).
Acknowledgements
We are thankful to the referee for their detailed report. We are also grateful to them for suggesting the current argument for Lemma 3.4. We are grateful to Jo Nelson and Jeremy Van Horn Morris for useful discussions. R. Casals is supported by the NSF Grant DMS-1841913 and a BBVA Research Fellowship. J. Etnyre is partially supported by the NSF Grant DMS-1608684.
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Casals, R., Etnyre, J.B. Non-simplicity of Isocontact Embeddings in All Higher Dimensions. Geom. Funct. Anal. 30, 1–33 (2020). https://doi.org/10.1007/s00039-020-00527-3
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DOI: https://doi.org/10.1007/s00039-020-00527-3