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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 ֒→ M which are displaceable through Hamiltonian isotopies (this happens for instance when M = 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. We also give topological characterisations of Lagrangians L ֒→ M with maximal Maslov number: when N L = dim(L) + 1 then L is homeomorphic to a sphere; when N L = n ≥ 6 then L fibers over the circle and the fiber is homeomorphic to a sphere. A consequence is that any exact Lagrangian in T * S 2k+1 whose Maslov class is zero is homeomorphic to S 2k+1 .Mathematics subject classification: 57R17, 57R58, 57R70, 53D12.

Abstract. We give topological obstructions to the existence of a closed exact Lagrangian submanifold L ,! T M , where M is the total space of a fibration over the circle. For instance, we show that 1 .L/ cannot be the free product of two non-trivial groups and that the difference between the number of generators and the number of relations in a finite presentation of 1 .L/ is less than two.

This paper provides examples of closed manifolds having a Morse number different from the stable Morse number. 2000 Mathematics Subject Classification 57Q10, 20F05.

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