Floer homology on the universal cover, a proof of Audin's conjecture and other constraints on Lagrangian submanifolds

On the topology of monotone Lagrangian submanifolds

“…For a generic choice of J, both M [g] and N [g] are closed manifolds of dimensions n + 2 resp. n (see [11]). There is a natural evaluation map…”

“…("stable fibration" instead of fibration), but only under a hypothesis related to the holomorphic disks of Maslov index equal to 2, with boundary in L (in particular the Maslov number of these submanifolds is supposed to be N L = 2). Note that a topological constraint for monotone Lagrangians with N L = n was established in our previous paper ( [11], Th.1.7). Recall that the Maslov number N L ∈ N is defined to be the positive generator of Im(I µ ).…”

“…If L ⊂ M is a closed monotone orientable Lagrangian which is displaceable and satisfies the condition (a) of Th. 1.3, then it was shown in [11] (Theorem 3.1 below) that N L = 2 and # g 2 = 0 for some g ∈ π 1 (L). But this has no direct consequence on the vanishing of # 2 .…”

“…In [11] we developed a new version of Floer homology theory for monotone Lagrangian submanifolds which involves some (arbitrary) covering space of the given Lagrangian. The outcome of this theory when L ⊂ M is monotone, displaceable with N L ≥ 3 is a spectral sequence whose initial page is built with the homology of the considered covering space and whose limit is zero.…”

“…In dimension three we get as a corollary that the only orientable Lagrangians in C 3 are products S 1 × Σ. The main ingredient of our proofs is the lifted Floer homology theory which we developed in [11].…”