Obstructions to the existence of monotone Lagrangian embeddings into cotangent bundles of manifolds fibered over the circle
Annales de l'Institut Fourier, Volume 59 (2009) no. 3, p. 1135-1175

We extend the constructions and results of Damian to get topological obstructions to the existence of closed monotone Lagrangian embeddings into the cotangent bundle of a space which is the total space of a fibration over the circle.

On étend des constructions et des résultats obtenus par Damian afin d’obtenir des obstructions topologiques à l’existence de plongements lagrangiens monotones dans le fibré cotangent d’un espace fibré sur le cercle.

DOI : https://doi.org/10.5802/aif.2460
Classification:  57R17,  57R58,  57R70,  53D12
Keywords: Lagrangian embeddings, Floer homology, Novikov homology.
@article{AIF_2009__59_3_1135_0,
     author = {Gadbled, Agn\`es},
     title = {Obstructions to the existence of monotone Lagrangian embeddings into cotangent bundles of manifolds fibered over the circle},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {59},
     number = {3},
     year = {2009},
     pages = {1135-1175},
     doi = {10.5802/aif.2460},
     mrnumber = {2543665},
     zbl = {1186.57019},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2009__59_3_1135_0}
}
Gadbled, Agnès. Obstructions to the existence of monotone Lagrangian embeddings into cotangent bundles of manifolds fibered over the circle. Annales de l'Institut Fourier, Volume 59 (2009) no. 3, pp. 1135-1175. doi : 10.5802/aif.2460. http://www.numdam.org/item/AIF_2009__59_3_1135_0/

[1] ArnolʼD, V. I. The first steps of symplectic topology, Uspekhi Mat. Nauk, Tome 41 (1986) no. 6 (252), p. 3-18, 229 | MR 890489 | Zbl 0618.58021

[2] Audin, Michèle Fibrés normaux d’immersions en dimension double, points doubles d’immersions lagrangiennes et plongements totalement réels, Comment. Math. Helv., Tome 63 (1988) no. 4, pp. 593-623 | Article | MR 966952 | Zbl 0666.57024

[3] Biran, Paul Lagrangian non-intersections, Geom. Funct. Anal., Tome 16 (2006) no. 2, pp. 279-326 | Article | MR 2231465 | Zbl 1099.53054

[4] Buhovski, Lev Homology of Lagrangian Submanifolds in Cotangent Bundles (arXiv:math/0312265) | Zbl 1090.53063

[5] Damian, Mihai Constraints on exact Lagrangians in cotangent bundles of manifolds fibered over the circle (arXiv:0710.0511, to be published in Comment. Math. Helv.)

[6] Floer, Andreas Morse theory for Lagrangian intersections, J. Differential Geom., Tome 28 (1988) no. 3, pp. 513-547 | MR 965228 | Zbl 0674.57027

[7] Floer, Andreas The unregularized gradient flow of the symplectic action, Comm. Pure Appl. Math., Tome 41 (1988) no. 6, pp. 775-813 | Article | MR 948771 | Zbl 0633.53058

[8] Fukaya, Kenji; Seidel, Paul; Smith, Ivan Exact Lagrangian submanifolds in simply-connected cotangent bundles, Invent. Math., Tome 172 (2008) no. 1, pp. 1-27 | Article | MR 2385665 | Zbl 1140.53036

[9] Hind, Richard Lagrangian spheres in S 2 ×S 2 , Geom. Funct. Anal., Tome 14 (2004) no. 2, pp. 303-318 | Article | MR 2060197 | Zbl 1066.53129

[10] Lalonde, François; Sikorav, Jean-Claude Sous-variétés lagrangiennes et lagrangiennes exactes des fibrés cotangents, Comment. Math. Helv., Tome 66 (1991) no. 1, pp. 18-33 | Article | MR 1090163 | Zbl 0759.53022

[11] Latour, François Existence de 1-formes fermées non singulières dans une classe de cohomologie de de Rham, Inst. Hautes Études Sci. Publ. Math. (1994) no. 80, pp. 135-194 | Article | Numdam | MR 1320607 | Zbl 0837.58002

[12] Lê, Hông Vân; Ono, Kaoru Symplectic fixed points, the Calabi invariant and Novikov homology, Topology, Tome 34 (1995) no. 1, pp. 155-176 | Article | MR 1308493 | Zbl 0822.58019

[13] Levitt, Gilbert 1-formes fermées singulières et groupe fondamental, Invent. Math., Tome 88 (1987) no. 3, pp. 635-667 | Article | MR 884804 | Zbl 0594.57014

[14] Nadler, David Microlocal branes are constructible sheaves (arXiv:math/0612399) | Zbl 1197.53116

[15] Novikov, S. P. Multivalued functions and functionals. An analogue of the Morse theory, Dokl. Akad. Nauk SSSR, Tome 260 (1981) no. 1, pp. 31-35 | MR 630459 | Zbl 0505.58011

[16] Oh, Yong-Geun Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I, Comm. Pure Appl. Math., Tome 46 (1993) no. 7, pp. 949-993 | Article | MR 1223659 | Zbl 0795.58019

[17] Oh, Yong-Geun Addendum to: Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I., Comm. Pure Appl. Math., Tome 48 (1995) no. 11, pp. 1299-1302 | Article | MR 1367384 | Zbl 0847.58036

[18] Oh, Yong-Geun Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings, Internat. Math. Res. Notices (1996) no. 7, pp. 305-346 | Article | MR 1389956 | Zbl 0858.58017

[19] Pazhitnov, A. V. Surgery on the Novikov complex, K-Theory, Tome 10 (1996) no. 4, pp. 323-412 | Article | MR 1404410 | Zbl 0861.57042

[20] Polterovich, Leonid Monotone Lagrange submanifolds of linear spaces and the Maslov class in cotangent bundles, Math. Z., Tome 207 (1991) no. 2, pp. 217-222 | Article | MR 1109663 | Zbl 0703.58018

[21] Ritter, Alexander F. Novikov-symplectic cohomology and exact Lagrangian embeddings (arXiv:0711.1396) | Zbl 1175.57019 | Zbl pre05530196

[22] Sikorav, Jean-Claude Homologie de Novikov associée à une classe de cohomologie réelle de degré 1 , Thèse Orsay (1987) (Ph. D. Thesis) | MR 826424

[23] Viterbo, Claude Intersection de sous-variétés lagrangiennes, fonctionnelles d’action et indice des systèmes hamiltoniens, Bull. Soc. Math. France, Tome 115 (1987) no. 3, pp. 361-390 | Numdam | MR 926533 | Zbl 0639.58018

[24] Viterbo, Claude Functors and computations in Floer homology with applications. I, Geom. Funct. Anal., Tome 9 (1999) no. 5, pp. 985-1033 | Article | MR 1726235 | Zbl 0954.57015