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 $N_L=n\ge 6$ 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 $S^{2k+1}$.