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. https://www.numdam.org/articles/10.24033/asens.2243/

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