Unlinking and unknottedness of monotone Lagrangian submanifolds

Dimitroglou Rizell, Georgios¹ ; Evans, Jonathan David²

¹ Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, Bâtiment 425, F-91405 Orsay, France
² Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK

Geometry & topology, Tome 18 (2014) no. 2, pp. 997-1034.

Résumé

Under certain topological assumptions, we show that two monotone Lagrangian submanifolds embedded in the standard symplectic vector space with the same monotonicity constant cannot link one another and that, individually, their smooth knot type is determined entirely by the homotopy theoretic data which classifies the underlying Lagrangian immersion. The topological assumptions are satisfied by a large class of manifolds which are realised as monotone Lagrangians, including tori. After some additional homotopy theoretic calculations, we deduce that all monotone Lagrangian tori in the symplectic vector space of odd complex dimension at least five are smoothly isotopic.

Détail

DOI : 10.2140/gt.2014.18.997

Classification : 53D12
Keywords: Lagrangian submanifold, symplectic manifold, monotone, torus, knot

Bibliographie

[1] P Albers, On the extrinsic topology of Lagrangian submanifolds, Int. Math. Res. Not. (2005) 2341

[2] P Albers, Erratum for “On the extrinsic topology of Lagrangian submanifolds”, Int. Math. Res. Not. (2010) 1363

[3] V I Arnol’D, On a characteristic class entering into conditions of quantization, Funkcional. Anal. i Priložen. 1 (1967) 1

[4] M Audin, F Lalonde, L Polterovich, Symplectic rigidity: Lagrangian submanifolds, from: "Holomorphic curves in symplectic geometry" (editors M Audin, J Lafontaine), Progr. Math. 117, Birkhäuser (1994) 271

[5] P Biran, K Cieliebak, Lagrangian embeddings into subcritical Stein manifolds, Israel J. Math. 127 (2002) 221

[6] P Biran, O Cornea, A Lagrangian quantum homology, from: "New perspectives and challenges in symplectic field theory" (editors M Abreu, F Lalonde, L Polterovich), CRM Proc. Lecture Notes 49, Amer. Math. Soc. (2009) 1

[7] M S Borman, T J Li, W Wu, Spherical Lagrangians via ball packings and symplectic cutting,

[8] V Borrelli, New examples of Lagrangian rigidity, Israel J. Math. 125 (2001) 221

[9] R Bott, Lectures on Morse theory, old and new, Bull. Amer. Math. Soc. 7 (1982) 331

[10] F Bourgeois, A Morse–Bott approach to contact homology, from: "Symplectic and contact topology: Interactions and perspectives" (editors Y Eliashberg, B Khesin, F Lalonde), Fields Inst. Commun. 35, Amer. Math. Soc. (2003) 55

[11] F Bourgeois, Y Eliashberg, H Hofer, K Wysocki, E Zehnder, Compactness results in symplectic field theory, Geom. Topol. 7 (2003) 799

[12] L Buhovsky, The Maslov class of Lagrangian tori and quantum products in Floer cohomology, J. Topol. Anal. 2 (2010) 57

[13] Y Chekanov, F Schlenk, Notes on monotone Lagrangian twist tori, Electron. Res. Announc. Math. Sci. 17 (2010) 104

[14] K Cieliebak, U A Frauenfelder, A Floer homology for exact contact embeddings, Pacific J. Math. 239 (2009) 251

[15] K Cieliebak, K Mohnke, Compactness for punctured holomorphic curves, J. Symplectic Geom. 3 (2005) 589

[16] M Damian, Floer homology on the universal cover, Audin's conjecture and other constraints on Lagrangian submanifolds, Comment. Math. Helv. 87 (2012) 433

[17] D L Dragnev, Fredholm theory and transversality for noncompact pseudoholomorphic maps in symplectizations, Comm. Pure Appl. Math. 57 (2004) 726

[18] Y Eliashberg, L Polterovich, Unknottedness of Lagrangian surfaces in symplectic $4$–manifolds, Internat. Math. Res. Notices (1993) 295

[19] Y Eliashberg, L Polterovich, New applications of Luttinger's surgery, Comment. Math. Helv. 69 (1994) 512

[20] Y Eliashberg, L Polterovich, Local Lagrangian $2$–knots are trivial, Ann. of Math. 144 (1996) 61

[21] Y Eliashberg, L Polterovich, The problem of Lagrangian knots in four-manifolds, from: "Geometric topology" (editor W H Kazez), AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 313

[22] J D Evans, Lagrangian spheres in del Pezzo surfaces, J. Topol. 3 (2010) 181

[23] J D Evans, J Kędra, Remarks on monotone Lagrangians in $\mathbb{C}^n$,

[24] K Groh, M Schwarz, K Smoczyk, K Zehmisch, Mean curvature flow of monotone Lagrangian submanifolds, Math. Z. 257 (2007) 295

[25] M L Gromov, A topological technique for the construction of solutions of differential equations and inequalities, from: "Actes du Congrès International des Mathématiciens, tome 2", Gauthier-Villars (1971) 221

[26] A Haefliger, M W Hirsch, On the existence and classification of differentiable embeddings, Topology 2 (1963) 129

[27] R Hind, Lagrangian spheres in $S^2\times S^2$, Geom. Funct. Anal. 14 (2004) 303

[28] R Hind, Lagrangian unknottedness in Stein surfaces, Asian J. Math. 16 (2012) 1

[29] R Hind, A Ivrii, Isotopies of high genus Lagrangian surfaces,

[30] M A Kervaire, Some nonstable homotopy groups of Lie groups, Illinois J. Math. 4 (1960) 161

[31] L Lazzarini, Existence of a somewhere injective pseudo-holomorphic disc, Geom. Funct. Anal. 10 (2000) 829

[32] L Lazzarini, Relative frames on $J$–holomorphic curves, J. Fixed Point Theory Appl. 9 (2011) 213

[33] J A Lees, On the classification of Lagrange immersions, Duke Math. J. 43 (1976) 217

[34] T J Li, W Wu, Lagrangian spheres, symplectic surfaces and the symplectic mapping class group, Geom. Topol. 16 (2012) 1121

[35] K M Luttinger, Lagrangian tori in $\mathbf{R}^4$, J. Differential Geom. 42 (1995) 220

[36] L A Lyusternik, A I Fet, Variational problems on closed manifolds, Doklady Akad. Nauk SSSR 81 (1951) 17

[37] Y G Oh, Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks I, Comm. Pure Appl. Math. 46 (1993) 949

[38] L Polterovich, The surgery of Lagrange submanifolds, Geom. Funct. Anal. 1 (1991) 198

[39] V V Prasolov, Elements of homology theory, Graduate Studies in Mathematics 81, Amer. Math. Soc. (2007)

[40] J Robbin, D Salamon, The Maslov index for paths, Topology 32 (1993) 827

[41] R Siefring, Relative asymptotic behavior of pseudoholomorphic half-cylinders, Comm. Pure Appl. Math. 61 (2008) 1631