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Commentarii Mathematici Helvetici

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Floer homology on the universal cover, Audin’s conjecture and other constraints on Lagrangian submanifolds

Mihai Damian (1)

(1) Institut de Recherche Mathématique Avancée, Université de Strasbourg, 7 Rue René Descartes, 67084, STRASBOURG CEDEX, FRANCE

We establish a new version of Floer homology for monotone Lagrangian embeddings in symplectic manifolds. As applications, we get assertions for monotone Lagrangian submanifolds $L\hookrightarrow M$ which are displaceable through Hamiltonian isotopies (this happens for instance when $M=\mathbb{C}^{n}$). We show that when $L$ is aspherical, or more generally when the homology of its universal cover vanishes in odd degrees, its Maslov number $N_{L}$ equals 2. This is a generalization of Audin’s conjecture. We also give topological characterisations of Lagrangians $L\hookrightarrow M$ with maximal Maslov number: when $N_{L}=\dim (L)+1$ then $L$ is homeomorphic to a sphere; when $N_{L}=n\geq 6$ then $L$ fibers over the circle and the fiber is homeomorphic to a sphere. A consequence is that any exact Lagrangian in $T^{\ast}S^{2k+1}$ whose Maslov class is zero is homeomorphic to $S^{2k+1}$.

Keywords: Lagrangian embeddings, Floer homology, Maslov number