## Unlinking and unknottedness of monotone Lagrangian submanifolds.(English)Zbl 1311.53065

Summary: 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.

### MSC:

 53D12 Lagrangian submanifolds; Maslov index 57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)

### Keywords:

Lagrangian submanifold; symplectic manifold; monotone; torus; knot
Full Text:

### References:

 [1] P Albers, On the extrinsic topology of Lagrangian submanifolds, Int. Math. Res. Not. (2005) 2341 · Zbl 1126.53053 [2] P Albers, Erratum for “On the extrinsic topology of Lagrangian submanifolds”, Int. Math. Res. Not. (2010) 1363 · Zbl 1190.53076 [3] V I Arnol’d, On a characteristic class entering into conditions of quantization, Funkcional. Anal. i Prilo\vzen. 1 (1967) 1 · Zbl 0175.20303 [4] M Audin, F Lalonde, L Polterovich, Symplectic rigidity: Lagrangian submanifolds (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 · Zbl 1165.53378 [6] P Biran, O Cornea, A Lagrangian quantum homology (editors M Abreu, F Lalonde, L Polterovich), CRM Proc. Lecture Notes 49, Amer. Math. Soc. (2009) 1 · Zbl 1185.53087 [7] M S Borman, T J Li, W Wu, Spherical Lagrangians via ball packings and symplectic cutting, · Zbl 1285.53074 [8] V Borrelli, New examples of Lagrangian rigidity, Israel J. Math. 125 (2001) 221 · Zbl 1021.53056 [9] R Bott, Lectures on Morse theory, old and new, Bull. Amer. Math. Soc. 7 (1982) 331 · Zbl 0505.58001 [10] F Bourgeois, A Morse-Bott approach to contact homology (editors Y Eliashberg, B Khesin, F Lalonde), Fields Inst. Commun. 35, Amer. Math. Soc. (2003) 55 · Zbl 1046.57017 [11] F Bourgeois, Y Eliashberg, H Hofer, K Wysocki, E Zehnder, Compactness results in symplectic field theory, Geom. Topol. 7 (2003) 799 · Zbl 1131.53312 [12] L Buhovsky, The Maslov class of Lagrangian tori and quantum products in Floer cohomology, J. Topol. Anal. 2 (2010) 57 · Zbl 1235.53083 [13] Y Chekanov, F Schlenk, Notes on monotone Lagrangian twist tori, Electron. Res. Announc. Math. Sci. 17 (2010) 104 · Zbl 1201.53083 [14] K Cieliebak, U A Frauenfelder, A Floer homology for exact contact embeddings, Pacific J. Math. 239 (2009) 251 · Zbl 1221.53112 [15] K Cieliebak, K Mohnke, Compactness for punctured holomorphic curves, J. Symplectic Geom. 3 (2005) 589 · Zbl 1113.53053 [16] M Damian, Floer homology on the universal cover, Audin’s conjecture and other constraints on Lagrangian submanifolds, Comment. Math. Helv. 87 (2012) 433 · Zbl 1251.53049 [17] D L Dragnev, Fredholm theory and transversality for noncompact pseudoholomorphic maps in symplectizations, Comm. Pure Appl. Math. 57 (2004) 726 · Zbl 1063.53086 [18] Y Eliashberg, L Polterovich, Unknottedness of Lagrangian surfaces in symplectic $$4$$-manifolds, Internat. Math. Res. Notices (1993) 295 · Zbl 0808.57021 [19] Y Eliashberg, L Polterovich, New applications of Luttinger’s surgery, Comment. Math. Helv. 69 (1994) 512 · Zbl 0853.57012 [20] Y Eliashberg, L Polterovich, Local Lagrangian $$2$$-knots are trivial, Ann. of Math. 144 (1996) 61 · Zbl 0872.57030 [21] Y Eliashberg, L Polterovich, The problem of Lagrangian knots in four-manifolds (editor W H Kazez), AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 313 · Zbl 0889.57036 [22] J D Evans, Lagrangian spheres in del Pezzo surfaces, J. Topol. 3 (2010) 181 · Zbl 1235.53084 [23] J D Evans, J K\cedra, Remarks on monotone Lagrangians in $$\mathbbC^n$$, [24] K Groh, M Schwarz, K Smoczyk, K Zehmisch, Mean curvature flow of monotone Lagrangian submanifolds, Math. Z. 257 (2007) 295 · Zbl 1144.53084 [25] M L Gromov, A topological technique for the construction of solutions of differential equations and inequalities, Gauthier-Villars (1971) 221 · Zbl 0237.57019 [26] A Haefliger, M W Hirsch, On the existence and classification of differentiable embeddings, Topology 2 (1963) 129 · Zbl 0113.38607 [27] R Hind, Lagrangian spheres in $$S^2\times S^2$$, Geom. Funct. Anal. 14 (2004) 303 · Zbl 1066.53129 [28] R Hind, Lagrangian unknottedness in Stein surfaces, Asian J. Math. 16 (2012) 1 · Zbl 1262.53073 [29] R Hind, A Ivrii, Isotopies of high genus Lagrangian surfaces, · Zbl 1200.57017 [30] M A Kervaire, Some nonstable homotopy groups of Lie groups, Illinois J. Math. 4 (1960) 161 · Zbl 0105.35302 [31] L Lazzarini, Existence of a somewhere injective pseudo-holomorphic disc, Geom. Funct. Anal. 10 (2000) 829 · Zbl 1003.32004 [32] L Lazzarini, Relative frames on $$J$$-holomorphic curves, J. Fixed Point Theory Appl. 9 (2011) 213 · Zbl 1236.57035 [33] J A Lees, On the classification of Lagrange immersions, Duke Math. J. 43 (1976) 217 · Zbl 0329.58006 [34] T J Li, W Wu, Lagrangian spheres, symplectic surfaces and the symplectic mapping class group, Geom. Topol. 16 (2012) 1121 · Zbl 1253.53073 [35] K M Luttinger, Lagrangian tori in $$\mathbfR^4$$, J. Differential Geom. 42 (1995) 220 · Zbl 0861.53029 [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 · Zbl 0795.58019 [38] L Polterovich, The surgery of Lagrange submanifolds, Geom. Funct. Anal. 1 (1991) 198 · Zbl 0754.57027 [39] V V Prasolov, Elements of homology theory, Graduate Studies in Mathematics 81, Amer. Math. Soc. (2007) · Zbl 1120.55001 [40] J Robbin, D Salamon, The Maslov index for paths, Topology 32 (1993) 827 · Zbl 0798.58018 [41] R Siefring, Relative asymptotic behavior of pseudoholomorphic half-cylinders, Comm. Pure Appl. Math. 61 (2008) 1631 · Zbl 1158.53068
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