We establish a new version of Floer homology for monotone Lagrangian embeddings in symplectic manifolds. As applications, we get assertions for monotone Lagrangian submanifolds which are displaceable through Hamiltonian isotopies (this happens for instance when ). We show that when is aspherical, or more generally when the homology of its universal cover vanishes in odd degrees, its Maslov number equals 2. This is a generalization of Audin’s conjecture. We also give topological characterisations of Lagrangians with maximal Maslov number: when then is homeomorphic to a sphere; when then fibers over the circle and the fiber is homeomorphic to a sphere. A consequence is that any exact Lagrangian in whose Maslov class is zero is homeomorphic to .
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. 433–463DOI 10.4171/CMH/259