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.
Accepted:
Published online:
Keywords: Legendrian, Floer, Positive isotopies
Mot clés : Legendrienne, Floer, Isotopies positives
Chantraine, Baptiste 1; Colin, Vincent 1; Dimitroglou Rizell, Georgios 2
@article{AIF_2019__69_4_1679_0, author = {Chantraine, Baptiste and Colin, Vincent and Dimitroglou Rizell, Georgios}, title = {Positive {Legendrian} isotopies and {Floer} theory}, journal = {Annales de l'Institut Fourier}, pages = {1679--1737}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {69}, number = {4}, year = {2019}, doi = {10.5802/aif.3279}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3279/} }
TY - JOUR AU - Chantraine, Baptiste AU - Colin, Vincent AU - Dimitroglou Rizell, Georgios TI - Positive Legendrian isotopies and Floer theory JO - Annales de l'Institut Fourier PY - 2019 SP - 1679 EP - 1737 VL - 69 IS - 4 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3279/ DO - 10.5802/aif.3279 LA - en ID - AIF_2019__69_4_1679_0 ER -
%0 Journal Article %A Chantraine, Baptiste %A Colin, Vincent %A Dimitroglou Rizell, Georgios %T Positive Legendrian isotopies and Floer theory %J Annales de l'Institut Fourier %D 2019 %P 1679-1737 %V 69 %N 4 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3279/ %R 10.5802/aif.3279 %G en %F AIF_2019__69_4_1679_0
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] An introduction to compactness results in symplectic field theory, Springer, 2014, viii+252 pages | DOI | MR | Zbl
[2] On the Floer homology of cotangent bundles, Commun. Pure Appl. Math., Volume 59 (2006) no. 2, pp. 254-316 Corrigendum in ibid. 67 (2014), no. 4, p. 670–691 | DOI | MR | Zbl
[3] An open string analogue of Viterbo functoriality, Geom. Topol., Volume 14 (2010) no. 2, pp. 627-718 | DOI | MR | Zbl
[4] Immersed Lagrangian Floer theory, J. Differ. Geom., Volume 86 (2010) no. 3, pp. 381-500 | MR | Zbl
[5] Orderability and the Weinstein conjecture, Compos. Math., Volume 151 (2015) no. 12, pp. 2251-2272 | DOI | MR | Zbl
[6] Orderability, contact non-squeezing, and Rabinowitz Floer homology, J. Symplectic Geom., Volume 16 (2018) no. 6, pp. 1481-1547 | DOI | MR | Zbl
[7] Semi-global Kuranishi charts and the definition of contact homology (2015) (https://arxiv.org/abs/1512.00580) | Zbl
[8] Existence and classification of overtwisted contact structures in all dimensions, Acta Math., Volume 215 (2015) no. 2, pp. 281-361 | DOI | MR | Zbl
[9] Effect of Legendrian surgery, Geom. Topol., Volume 16 (2012) no. 1, pp. 301-389 | DOI | MR | Zbl
[10] Compactness results in symplectic field theory, Geom. Topol., Volume 7 (2003), pp. 799-888 | DOI | MR | Zbl
[11] Lagrangian concordance of Legendrian knots, Algebr. Geom. Topol., Volume 10 (2010) no. 1, pp. 63-85 | DOI | MR | Zbl
[12] Floer homology and Lagrangian concordance, Proceedings of 21st Gökova Geometry-Topology Conference 2014, Gökova Geometry/Topology Conference (GGT), 2015, pp. 76-113 | Zbl
[13] Floer theory for Lagrangian cobordisms (2015) (https://arxiv.org/abs/1511.09471) | Zbl
[14] Differential algebra of Legendrian links, Invent. Math., Volume 150 (2002) no. 3, pp. 441-483 | DOI | MR | Zbl
[15] Non-negative Legendrian isotopy in , Geom. Topol., Volume 14 (2010) no. 1, pp. 611-626 | DOI | MR | Zbl
[16] Universal orderability of Legendrian isotopy classes, J. Symplectic Geom., Volume 14 (2016) no. 1, pp. 149-170 | DOI | MR | Zbl
[17] Symplectic homology and the Eilenberg–Steenrod axioms, Algebr. Geom. Topol., Volume 18 (2018) no. 4, pp. 1953-2130 | DOI | MR | Zbl
[18] Positive isotopies of Legendrian submanifolds and applications, Int. Math. Res. Not., Volume 2017 (2017) no. 20, pp. 6231-6254 | DOI | MR | Zbl
[19] The discriminant and oscillation lengths for contact and Legendrian isotopies, J. Eur. Math. Soc., Volume 17 (2015) no. 7, pp. 1657-1685 | DOI | MR | Zbl
[20] The Bott-Samelson theorem for positive Legendrian isotopies, Abh. Math. Semin. Univ. Hamb., Volume 88 (2018) no. 1, pp. 87-96 | DOI | MR | Zbl
[21] Lifting pseudo-holomorphic polygons to the symplectisation of and applications, Quantum Topol., Volume 7 (2016) no. 1, pp. 29-105 | DOI | MR | Zbl
[22] An energy-capacity inequality for Legendrian submanifolds (2016) (https://arxiv.org/abs/1608.06232) | Zbl
[23] Morse flow trees and Legendrian contact homology in 1-jet spaces, Geom. Topol., Volume 11 (2007), pp. 1083-1224 | DOI | MR | Zbl
[24] Rational symplectic field theory over for exact Lagrangian cobordisms, J. Eur. Math. Soc., Volume 10 (2008) no. 3, pp. 641-704 | DOI | MR | Zbl
[25] Rational SFT, linearized Legendrian contact homology, and Lagrangian Floer cohomology, Perspectives in analysis, geometry, and topology (Progress in Mathematics), Volume 296, Birkhäuser, 2012, pp. 109-145 | DOI | MR | Zbl
[26] A duality exact sequence for Legendrian contact homology, Duke Math. J., Volume 150 (2009) no. 1, pp. 1-75 | DOI | MR | Zbl
[27] Legendrian contact homology in , Trans. Am. Math. Soc., Volume 359 (2007) no. 7, pp. 3301-3335 | DOI | MR | Zbl
[28] Legendrian knots and exact Lagrangian cobordisms, J. Eur. Math. Soc., Volume 18 (2016) no. 11, pp. 2627-2689 | DOI | MR | Zbl
[29] 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] Lagrangian intersection theory: finite-dimensional approach, Geometry of differential equations (Translations. Series 2), Volume 186, American Mathematical Society, 1998, pp. 27-118 | MR | Zbl
[31] Lagrangian intersections in contact geometry, Geom. Funct. Anal., Volume 5 (1995) no. 2, pp. 244-269 | DOI | MR | Zbl
[32] Geometry of contact transformations and domains: orderability versus squeezing, Geom. Topol., Volume 10 (2006), pp. 1635-1747 | DOI | MR | Zbl
[33] Partially ordered groups and geometry of contact transformations, Geom. Funct. Anal., Volume 10 (2000) no. 6, pp. 1448-1476 | DOI | MR | Zbl
[34] Morse theory for Lagrangian intersections, J. Differ. Geom., Volume 28 (1988) no. 3, pp. 513-547 | MR | Zbl
[35] On Sandon-type metrics for contactomorphism groups, Ann. Math. Qué. (2017), pp. 191-214 | DOI | Zbl
[36] Lagrangian intersection Floer theory: anomaly and obstruction. Part I, AMS/IP Studies in Advanced Mathematics, 46, American Mathematical Society; International Press, 2009, xii+396 pages | MR | Zbl
[37] Nonlinear generalization of the Maslov index, Theory of singularities and its applications (Advances in Soviet Mathematics), Volume 1, American Mathematical Society, 1990, pp. 71-103 | DOI | MR | Zbl
[38] Sheaf quantization of Hamiltonian isotopies and applications to nondisplaceability problems, Duke Math. J., Volume 161 (2012) no. 2, pp. 201-245 | DOI | MR | Zbl
[39] On positive loops of loose Legendrian embeddings, Université de Nantes (France) (2016) (Ph. D. Thesis)
[40] On positive loops of loose Legendrian embeddings (2016) (https://arxiv.org/abs/1605.07494)
[41] La suite spectrale de Leray–Serre en cohomologie de Floer pour variétés sympelctiques compactes à bor de type contact, Université Paris XI (France) (2003) (Ph. D. Thesis)
[42] Symplectic topology as the geometry of action functional. I. Relative Floer theory on the cotangent bundle, J. Differ. Geom., Volume 46 (1997) no. 3, pp. 499-577 | MR | Zbl
[43] A simple construction of positive loops of Legendrians, Ark. Mat., Volume 56 (2018) no. 2, pp. 377-394 | DOI | MR | Zbl
[44] Contact homology and virtual fundamental cycles (2015) (https://arxiv.org/abs/1508.03873) | Zbl
[45] Topological quantum field theory structure on symplectic cohomology, J. Topol., Volume 6 (2013) no. 2, pp. 391-489 | DOI | MR | Zbl
[46] The minimal length of a Lagrangian cobordism between Legendrians, Sel. Math., New Ser., Volume 23 (2017) no. 2, pp. 1419-1448 | DOI | MR | Zbl
[47] An integer-valued bi-invariant metric on the group of contactomorphisms of , J. Topol. Anal., Volume 2 (2010) no. 3, pp. 327-339 | DOI | MR | Zbl
[48] Bi-invariant metrics on the contactomorphism groups, São Paulo J. Math. Sci., Volume 9 (2015) no. 2, pp. 195-228 | DOI | MR | Zbl
[49] Floer homology for translated points (2016) (In preparation) | Zbl
[50] Some properties of holomorphic curves in almost complex manifolds, Holomorphic curves in symplectic geometry (Progress in Mathematics), Volume 117, Birkhäuser, 1994, pp. 165-189 | DOI | MR | Zbl
[51] Symplectic topology as the geometry of generating functions, Math. Ann., Volume 292 (1992) no. 4, pp. 685-710 | DOI | MR | Zbl
[52] Geometry of Contactomorphism Groups, Contact Rigidity, and Contact Dynamics in Jet Spaces, Int. Math. Res. Not., Volume 2013 (2013) no. 20, pp. 4687-4711 | DOI | MR | Zbl
[53] Théorèmes de Künneth en homologie de contact, Université Libre de Bruxelles (Belgium) (2014) (Ph. D. Thesis)
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