Positive Legendrian isotopies and Floer theory
Annales de l'Institut Fourier, Volume 69 (2019) no. 4, pp. 1679-1737.

Positive loops of Legendrian embeddings are examined from the point of view of Floer homology of Lagrangian cobordisms. This leads to new obstructions to the existence of a positive loop containing a given Legendrian, expressed in terms of the Legendrian contact homology of the Legendrian submanifold. As applications, old and new examples of orderable contact manifolds are obtained and discussed. We also show that contact manifolds filled by a Liouville domain with non-zero symplectic homology are strongly orderable in the sense of Liu.

On étudie les lacets positifs de plongements legendriens du point de vue de l’homologie de Floer pour les cobordismes lagrangiens. On obtient ainsi de nouvelles obstructions à l’existence d’un lacet positif contenant une sous-variété legendrienne donnée, exprimées à l’aide de son homologie de contact legendrienne. On applique ensuite ces obstructions pour revisiter d’anciens et donner de nouveaux exemples de variétés de contact ordonnables. On démontre également qu’une variété de contact remplissable par un domaine de Liouville dont l’homologie symplectique est non triviale est fortement ordonnable au sens de Liu.

Received:
Accepted:
Published online:
DOI: 10.5802/aif.3279
Classification: 53D42,  57R58,  53D10
Keywords: Legendrian, Floer, Positive isotopies
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {Positive {Legendrian} isotopies and {Floer} theory},
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Chantraine, Baptiste; Colin, Vincent; Dimitroglou Rizell, Georgios. Positive Legendrian isotopies and Floer theory. Annales de l'Institut Fourier, Volume 69 (2019) no. 4, pp. 1679-1737. doi : 10.5802/aif.3279. https://aif.centre-mersenne.org/articles/10.5802/aif.3279/

[1] Abbas, Casim An introduction to compactness results in symplectic field theory, Springer, 2014, viii+252 pages | DOI | MR | Zbl

[2] Abbondandolo, Alberto; Schwarz, Matthias On the Floer homology of cotangent bundles, Commun. Pure Appl. Math., Tome 59 (2006) no. 2, pp. 254-316 (Corrigendum in ibid. 67 (2014), no. 4, p. 670–691) | DOI | MR | Zbl

[3] Abouzaid, Mohammed; Seidel, Paul An open string analogue of Viterbo functoriality, Geom. Topol., Tome 14 (2010) no. 2, pp. 627-718 | DOI | MR | Zbl

[4] Akaho, Manabu; Joyce, Dominic Immersed Lagrangian Floer theory, J. Differ. Geom., Tome 86 (2010) no. 3, pp. 381-500 | MR | Zbl

[5] Albers, Peter; Fuchs, Urs; Merry, Will J. Orderability and the Weinstein conjecture, Compos. Math., Tome 151 (2015) no. 12, pp. 2251-2272 | DOI | MR | Zbl

[6] Albers, Peter; Merry, Will J. Orderability, contact non-squeezing, and Rabinowitz Floer homology, J. Symplectic Geom., Tome 16 (2018) no. 6, pp. 1481-1547 | DOI | MR | Zbl

[7] Bao, Erkao; Honda, Ko Semi-global Kuranishi charts and the definition of contact homology (2015) (https://arxiv.org/abs/1512.00580)

[8] Borman, Matthew S.; Eliashberg, Yasha; Murphy, Emmy Existence and classification of overtwisted contact structures in all dimensions, Acta Math., Tome 215 (2015) no. 2, pp. 281-361 | DOI | MR | Zbl

[9] Bourgeois, Frédéric; Ekholm, Tobias; Eliashberg, Yasha Effect of Legendrian surgery, Geom. Topol., Tome 16 (2012) no. 1, pp. 301-389 | DOI | MR | Zbl

[10] Bourgeois, Frédéric; Eliashberg, Yasha; Hofer, Helmut; Wysocki, K.; Zehnder, E. Compactness results in symplectic field theory, Geom. Topol., Tome 7 (2003), pp. 799-888 | DOI | MR | Zbl

[11] Chantraine, Baptiste Lagrangian concordance of Legendrian knots, Algebr. Geom. Topol., Tome 10 (2010) no. 1, pp. 63-85 | DOI | MR | Zbl

[12] Chantraine, Baptiste; Dimitroglou Rizell, Georgios; Ghiggini, Paolo; Golovko, Roman Floer homology and Lagrangian concordance, Proceedings of 21st Gökova Geometry-Topology Conference 2014, Gökova Geometry/Topology Conference (GGT), 2015, pp. 76-113 | Zbl

[13] Chantraine, Baptiste; Dimitroglou Rizell, Georgios; Ghiggini, Paolo; Golovko, Roman Floer theory for Lagrangian cobordisms (2015) (https://arxiv.org/abs/1511.09471)

[14] Chekanov, Yuri V. Differential algebra of Legendrian links, Invent. Math., Tome 150 (2002) no. 3, pp. 441-483 | DOI | MR | Zbl

[15] Chernov, Vladimir; Nemirovski, Stefan Non-negative Legendrian isotopy in ST * M, Geom. Topol., Tome 14 (2010) no. 1, pp. 611-626 | DOI | MR | Zbl

[16] Chernov, Vladimir; Nemirovski, Stefan Universal orderability of Legendrian isotopy classes, J. Symplectic Geom., Tome 14 (2016) no. 1, pp. 149-170 | DOI | MR | Zbl

[17] Cieliebak, Kai; Oancea, Alexandru Symplectic homology and the Eilenberg–Steenrod axioms, Algebr. Geom. Topol., Tome 18 (2018) no. 4, pp. 1953-2130 | DOI | MR | Zbl

[18] Colin, Vincent; Ferrand, Emmanuel; Pushkar, Petya Positive isotopies of Legendrian submanifolds and applications, Int. Math. Res. Not., Tome 2017 (2017) no. 20, pp. 6231-6254 | DOI | MR | Zbl

[19] Colin, Vincent; Sandon, Sheila The discriminant and oscillation lengths for contact and Legendrian isotopies, J. Eur. Math. Soc., Tome 17 (2015) no. 7, pp. 1657-1685 | DOI | MR | Zbl

[20] Dahinden, Lucas The Bott-Samelson theorem for positive Legendrian isotopies, Abh. Math. Semin. Univ. Hamb., Tome 88 (2018) no. 1, pp. 87-96 | DOI | MR | Zbl

[21] Dimitroglou Rizell, Georgios Lifting pseudo-holomorphic polygons to the symplectisation of P× and applications, Quantum Topol., Tome 7 (2016) no. 1, pp. 29-105 | DOI | MR | Zbl

[22] Dimitroglou Rizell, Georgios; Sullivan, Michael An energy-capacity inequality for Legendrian submanifolds (2016) (https://arxiv.org/abs/1608.06232)

[23] Ekholm, Tobias Morse flow trees and Legendrian contact homology in 1-jet spaces, Geom. Topol., Tome 11 (2007), pp. 1083-1224 | DOI | MR | Zbl

[24] Ekholm, Tobias Rational symplectic field theory over Z 2 for exact Lagrangian cobordisms, J. Eur. Math. Soc., Tome 10 (2008) no. 3, pp. 641-704 | DOI | MR | Zbl

[25] Ekholm, Tobias Rational SFT, linearized Legendrian contact homology, and Lagrangian Floer cohomology, Perspectives in analysis, geometry, and topology (Progress in Mathematics) Tome 296, Birkhäuser, 2012, pp. 109-145 | DOI | MR | Zbl

[26] Ekholm, Tobias; Etnyre, John B.; Sabloff, Joshua M. A duality exact sequence for Legendrian contact homology, Duke Math. J., Tome 150 (2009) no. 1, pp. 1-75 | DOI | MR | Zbl

[27] Ekholm, Tobias; Etnyre, John B.; Sullivan, Michael Legendrian contact homology in P×, Trans. Am. Math. Soc., Tome 359 (2007) no. 7, pp. 3301-3335 | DOI | MR | Zbl

[28] Ekholm, Tobias; Honda, Ko; Kálmán, Tamás Legendrian knots and exact Lagrangian cobordisms, J. Eur. Math. Soc., Tome 18 (2016) no. 11, pp. 2627-2689 | DOI | MR | Zbl

[29] Eliashberg, Yasha; Givental, Alexander; Hofer, Helmut Introduction to symplectic field theory, Geom. Funct. Anal. (2000) no. Special Volume, Part II, pp. 560-673 (GAFA 2000 (Tel Aviv, 1999)) | MR | Zbl

[30] Eliashberg, Yasha; Gromov, Misha Lagrangian intersection theory: finite-dimensional approach, Geometry of differential equations (Translations. Series 2) Tome 186, American Mathematical Society, 1998, pp. 27-118 | MR | Zbl

[31] Eliashberg, Yasha; Hofer, Helmut; Salamon, Dietmar Lagrangian intersections in contact geometry, Geom. Funct. Anal., Tome 5 (1995) no. 2, pp. 244-269 | DOI | MR | Zbl

[32] Eliashberg, Yasha; Kim, Sang Seon; Polterovich, Leonid Geometry of contact transformations and domains: orderability versus squeezing, Geom. Topol., Tome 10 (2006), pp. 1635-1747 | DOI | MR | Zbl

[33] Eliashberg, Yasha; Polterovich, Leonid Partially ordered groups and geometry of contact transformations, Geom. Funct. Anal., Tome 10 (2000) no. 6, pp. 1448-1476 | DOI | MR | Zbl

[34] Floer, Andreas Morse theory for Lagrangian intersections, J. Differ. Geom., Tome 28 (1988) no. 3, pp. 513-547 | MR | Zbl

[35] Fraser, Maia; Polterovich, Leonid; Rosen, Daniel On Sandon-type metrics for contactomorphism groups, Ann. Math. Qué. (2017), pp. 191-214 | DOI | Zbl

[36] Fukaya, Kenji; Oh, Yong-Geun; Ohta, Hiroshi; Ono, Kaoru Lagrangian intersection Floer theory: anomaly and obstruction. Part I, AMS/IP Studies in Advanced Mathematics, Tome 46, American Mathematical Society; International Press, 2009, xii+396 pages | MR | Zbl

[37] Givental, Alexander Nonlinear generalization of the Maslov index, Theory of singularities and its applications (Advances in Soviet Mathematics) Tome 1, American Mathematical Society, 1990, pp. 71-103 | DOI | MR | Zbl

[38] Guillermou, Stéphane; Kashiwara, Masaki; Schapira, Pierre Sheaf quantization of Hamiltonian isotopies and applications to nondisplaceability problems, Duke Math. J., Tome 161 (2012) no. 2, pp. 201-245 | DOI | MR | Zbl

[39] Liu, Guogang On positive loops of loose Legendrian embeddings (2016) (Ph. D. Thesis)

[40] Liu, Guogang On positive loops of loose Legendrian embeddings (2016) (https://arxiv.org/abs/1605.07494)

[41] Oancea, Alexandru La suite spectrale de Leray–Serre en cohomologie de Floer pour variétés sympelctiques compactes à bor de type contact (2003) (Ph. D. Thesis)

[42] Oh, Yong-Geun Symplectic topology as the geometry of action functional. I. Relative Floer theory on the cotangent bundle, J. Differ. Geom., Tome 46 (1997) no. 3, pp. 499-577 | MR | Zbl

[43] Pancholi, Dishant; Pérez, José L.; Presas, Francisco A simple construction of positive loops of Legendrians, Ark. Mat., Tome 56 (2018) no. 2, pp. 377-394 | DOI | MR | Zbl

[44] Pardon, John Contact homology and virtual fundamental cycles (2015) (https://arxiv.org/abs/1508.03873)

[45] Ritter, Alexander F. Topological quantum field theory structure on symplectic cohomology, J. Topol., Tome 6 (2013) no. 2, pp. 391-489 | DOI | MR | Zbl

[46] Sabloff, Joshua M.; Traynor, Lisa The minimal length of a Lagrangian cobordism between Legendrians, Sel. Math., New Ser., Tome 23 (2017) no. 2, pp. 1419-1448 | DOI | MR | Zbl

[47] Sandon, Sheila An integer-valued bi-invariant metric on the group of contactomorphisms of 2n ×S 1 , J. Topol. Anal., Tome 2 (2010) no. 3, pp. 327-339 | DOI | MR | Zbl

[48] Sandon, Sheila Bi-invariant metrics on the contactomorphism groups, São Paulo J. Math. Sci., Tome 9 (2015) no. 2, pp. 195-228 | DOI | MR | Zbl

[49] Sandon, Sheila Floer homology for translated points (2016) (In preparation)

[50] Sikorav, Jean-Claude Some properties of holomorphic curves in almost complex manifolds, Holomorphic curves in symplectic geometry (Progress in Mathematics) Tome 117, Birkhäuser, 1994, pp. 165-189 | DOI | MR | Zbl

[51] Viterbo, Claude Symplectic topology as the geometry of generating functions, Math. Ann., Tome 292 (1992) no. 4, pp. 685-710 | DOI | MR | Zbl

[52] Zapolsky, Frol Geometry of Contactomorphism Groups, Contact Rigidity, and Contact Dynamics in Jet Spaces, Int. Math. Res. Not., Tome 2013 (2013) no. 20, pp. 4687-4711 | DOI | MR | Zbl

[53] Zenaïdi, Naim Théorèmes de Künneth en homologie de contact (2014) (Ph. D. Thesis)

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