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

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 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} .