On the topology of monotone Lagrangian submanifolds
[Sur la topologie des sous-variétés lagrangiennes monotones]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 1, pp. 237-252.

Nous trouvons de nouvelles obstructions sur la topologie des sous-variétés lagrangiennes compactes monotones de 𝐂n sous certaines hypothèses sur l'homologie de leur revêtement universel. Nous montrons en particulier que les sommes connexes non-triviales de variétés compactes de dimension impaire n'admettent pas de plongement lagrangien monotone dans 𝐂n : la question de l'existence de tels plongements était ouverte. En dimension trois nous obtenons comme corollaire que les seules sous-variétés lagrangiennes compactes monototones et orientables de 𝐂3 sont les produits 𝐒1×Σ. L'outil principal de nos preuves est l'homologie de Floer relevée que nous avons définie en [13].

We find new obstructions on the topology of closed monotone Lagrangian submanifolds of 𝐂n under some hypotheses on the homology of their universal cover. In particular we show that nontrivial connected sums of manifolds of odd dimensions do not admit monotone Lagrangian embeddings into 𝐂n whereas some of these examples are known to admit usual Lagrangian embeddings: the question of the existence of a monotone embedding for a given Lagrangian in 𝐂n was open. In dimension three we get as a corollary that the only orientable Lagrangians in 𝐂3 are products 𝐒1×Σ. The main ingredient of our proofs is the lifted Floer homology theory which we developed in [13].

Publié le :
DOI : 10.24033/asens.2243
Classification : 57R17, 57R58, 57R70, 53D12.
Keywords: Monotone Lagrangian embeddings, Novikov homology, lifted Floer homology.
Mot clés : Plongements lagrangiens monotones, homologues de Novikow, homologie de Floer relevée. 
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     title = {On the topology  of monotone {Lagrangian} submanifolds},
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     pages = {237--252},
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     volume = {Ser. 4, 48},
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Damian, Mihai. On the topology  of monotone Lagrangian submanifolds. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 1, pp. 237-252. doi : 10.24033/asens.2243. http://www.numdam.org/articles/10.24033/asens.2243/

Arnolʼd, V. I. On a characteristic class entering into conditions of quantization, Funkcional. Anal. i Priložen., Volume 1 (1967), pp. 1-14 (ISSN: 0374-1990) | DOI | MR | Zbl

Audin, M. Fibrés normaux d'immersions en dimension double, points doubles d'immersions lagrangiennes et plongements totalement réels, Comment. Math. Helv., Volume 63 (1988), pp. 593-623 (ISSN: 0010-2571) | DOI | MR | Zbl

Baer, R. Isotopie von Kurven auf orientierbaren, geschlossenen Flächen und ihr Zusammenhang mit der topologischen Deformation der Flächen, J. reine angew. Math., Volume 159 (1928), pp. 101-111 | DOI | JFM | MR

Biran, P.; Cornea, O. Quantum Structures for Lagrangian submanifolds (preprint arXiv:0808.3989 )

Biran, P.; Cieliebak, K. Symplectic topology on subcritical manifolds, Comment. Math. Helv., Volume 76 (2001), pp. 712-753 (ISSN: 0010-2571) | DOI | MR | Zbl

Barraud, J.-F.; Cornea, O. Quantization of the Serre spectral sequence, J. Symplectic Geom., Volume 5 (2007), pp. 249-280 http://projecteuclid.org/euclid.jsg/1210083199 (ISSN: 1527-5256) | DOI | MR | Zbl

Biran, P.; Cornea, O., New perspectives and challenges in symplectic field theory (CRM Proc. Lecture Notes), Volume 49, Amer. Math. Soc., Providence, RI, 2009, pp. 1-44 | DOI | MR | Zbl

Biran, P.; Cornea, O. Rigidity and uniruling for Lagrangian submanifolds, Geom. Topol., Volume 13 (2009), pp. 2881-2989 (ISSN: 1465-3060) | DOI | MR | Zbl

Biran, P.; Cornea, O. Lagrangian topology and enumerative geometry, Geom. Topol., Volume 16 (2012), pp. 963-1052 (ISSN: 1465-3060) | DOI | MR | Zbl

Biran, P. Lagrangian non-intersections, Geom. Funct. Anal., Volume 16 (2006), pp. 279-326 (ISSN: 1016-443X) | DOI | MR | Zbl

Damian, M. Constraints on exact Lagrangians in cotangent bundles of manifolds fibered over the circle, Comment. Math. Helv., Volume 84 (2009), pp. 705-746 (ISSN: 0010-2571) | DOI | MR | Zbl

Damian, M. On the homotopy of finite CW-complexes with polycyclic fundamental group, Trans. Amer. Math. Soc., Volume 361 (2009), pp. 1791-1809 (ISSN: 0002-9947) | DOI | MR | Zbl

Damian, M. Floer homology on the universal cover, Audin's conjecture and other constraints on Lagrangian submanifolds, Comment. Math. Helv., Volume 87 (2012), pp. 433-462 (ISSN: 0010-2571) | DOI | MR | Zbl

Dehn, M. Die Gruppe der Abbildungsklassen, Acta Math., Volume 69 (1938), pp. 135-206 (ISSN: 0001-5962) | DOI | JFM | MR | Zbl

Ekholm, T.; Eliashberg, Y.; Murphy, E.; Smith, I. Constructing exact Lagrangian immersions with few double points, Geom. Funct. Anal., Volume 23 (2013), pp. 1772-1803 (ISSN: 1016-443X) | DOI | MR | Zbl

Evans, J. D.; Kedra, J. Remarks on monotone Lacrangians in 𝐂 n (preprint arXiv:1110.0927 ) | MR

Fukaya, K.; Oh, Y.-G.; Ohta, H.; Ono, K., AMS/IP Studies in Advanced Math., 46, 2009 | MR | Zbl

Fukaya, K., Morse theoretic methods in nonlinear analysis and in symplectic topology (NATO Sci. Ser. II Math. Phys. Chem.), Volume 217, Springer, Dordrecht, 2006, pp. 231-276 | DOI | MR | Zbl

Giventalʼ, A. B. Lagrangian imbeddings of surfaces and the open Whitney umbrella, Funktsional. Anal. i Prilozhen., Volume 20 (1986), pp. 35-41 (ISSN: 0374-1990) | DOI | MR | Zbl

Gromov, M. Pseudoholomorphic curves in symplectic manifolds, Invent. Math., Volume 82 (1985), pp. 307-347 (ISSN: 0020-9910) | DOI | MR | Zbl

Latour, F. Existence de 1-formes fermées non singulières dans une classe de cohomologie de de Rham, Publ. Math. I.H.É.S., Volume 80 (1994), pp. 135-194 (ISSN: 0073-8301) | DOI | Numdam | MR | Zbl

Nemirovskiĭ, S. Y. The homology class of a Lagrangian Klein bottle, Izv. Ross. Akad. Nauk Ser. Mat., Volume 73 (2009), pp. 37-48 ; English translation: Izvestiya: Mathematics 73 (2009), 689–698 (ISSN: 0373-2436) | DOI | MR | Zbl

Nielsen, J. Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen, Acta Math., Volume 50 (1927), pp. 189-358 (ISSN: 0001-5962) | DOI | JFM | MR

Oh, Y.-G. Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings, Int. Math. Res. Not., Volume 1996 (1996), pp. 305-346 (ISSN: 1073-7928) | DOI | MR | Zbl

Pazhitnov, A. V. Surgery on the Novikov complex, K -Theory, Volume 10 (1996), pp. 323-412 (ISSN: 0920-3036) | DOI | MR | Zbl

Polterovich, L. The surgery of Lagrange submanifolds, Geom. Funct. Anal., Volume 1 (1991), pp. 198-210 (ISSN: 1016-443X) | DOI | MR | Zbl

Shevchishin, V. V. Lagrangian embeddings of the Klein bottle and the combinatorial properties of mapping class groups, Izv. Ross. Akad. Nauk Ser. Mat., Volume 73 (2009), pp. 153-224 ; English translation: Izvestiya: Mathematics 73 (2009), 797–859 (ISSN: 0373-2436) | DOI | MR | Zbl

Stallings, J., Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 95-100 | MR | Zbl

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