JournalscmhVol. 87, No. 2pp. 433–463

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

  • Mihai Damian

    Université de Strasbourg, France
Floer homology on the universal cover,  Audin’s conjecture and other constraints  on Lagrangian submanifolds cover
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Abstract

We establish a new version of Floer homology for monotone Lagrangian embeddings in symplectic manifolds. As applications, we get assertions for monotone Lagrangian submanifolds LML\hookrightarrow M which are displaceable through Hamiltonian isotopies (this happens for instance when M=CnM=\mathbb{C}^{n}). We show that when LL is aspherical, or more generally when the homology of its universal cover vanishes in odd degrees, its Maslov number NLN_{L} equals 2. This is a generalization of Audin’s conjecture. We also give topological characterisations of Lagrangians LML\hookrightarrow M with maximal Maslov number: when NL=dim(L)+1N_{L}=\dim (L)+1 then LL is homeomorphic to a sphere; when NL=n6N_{L}=n\geq 6 then LL fibers over the circle and the fiber is homeomorphic to a sphere. A consequence is that any exact Lagrangian in TS2k+1T^{\ast}S^{2k+1} whose Maslov class is zero is homeomorphic to S2k+1S^{2k+1}.

Cite this article

Mihai Damian, Floer homology on the universal cover, Audin’s conjecture and other constraints on Lagrangian submanifolds. Comment. Math. Helv. 87 (2012), no. 2, pp. 433–463

DOI 10.4171/CMH/259