On the topology  of monotone Lagrangian submanifolds

Damian, Mihai

doi:10.24033/asens.2243

Geodesic

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On the topology of monotone Lagrangian submanifolds
[Sur la topologie des sous-variétés lagrangiennes monotones]

Damian, Mihai

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 1, pp. 237-252.

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Abstract (VO)
Résumé

We find new obstructions on the topology of closed monotone Lagrangian submanifolds of $𝐂^{n}$ 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 $𝐂^{n}$ 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 $𝐂^{n}$ was open. In dimension three we get as a corollary that the only orientable Lagrangians in $𝐂^{3}$ are products $𝐒^{1} \times Σ$ . The main ingredient of our proofs is the lifted Floer homology theory which we developed in [13].

Nous trouvons de nouvelles obstructions sur la topologie des sous-variétés lagrangiennes compactes monotones de $𝐂^{n}$ 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 $𝐂^{n}$ : 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 $𝐂^{3}$ sont les produits $𝐒^{1} \times Σ$ . L'outil principal de nos preuves est l'homologie de Floer relevée que nous avons définie en [13].

Publié le : 2015-05-27

MR Zbl

DOI : 10.24033/asens.2243

Classification : 57R17, 57R58, 57R70, 53D12.
Keywords: Monotone Lagrangian embeddings, Novikov homology, lifted Floer homology.
Mots-clés : Plongements lagrangiens monotones, homologues de Novikow, homologie de Floer relevée.

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     title = {On the topology  of monotone {Lagrangian} submanifolds},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {237--252},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     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://geodesic.mathdoc.fr/articles/10.24033/asens.2243/

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