Floer homology on the universal cover, Audin's conjecture and other constraints on Lagrangian submanifolds
Let \((M^{2n}, \omega)\) be a symplectic manifold. A submanifold \(L^n\) of \(M\) is called \textit{Lagrangian} if the restriction of \(\omega\) to \(L\) vanishes. A fundamental question of symplectic geometry involves understanding necessary conditions for a manifold to admit a Lagrangian embedding in a given \(M\). The paper under review gives partial answers in the case of \textit{monotone} Lagrangian embeddings which are displaceable through Hamiltonian isotopies (this happens for instance when \(M = \mathbb{C}^n\)). The main result is that, if such \(L\) is aspherical, or more generally if the homology of its universal cover vanishes in odd degrees, its Maslov number \(N_L\) equals 2. The author also gives topological characterisations of Lagrangians \(L \hookrightarrow M\) with maximal Maslov number, in particular showing that if \(N_L = \dim(L) + 1\) then \(L\) is homeomorphic to a sphere, and if \(N_L = n \geq 6\) then \(L\) fibers over the circle and the fiber is homeomorphic to a sphere. A consequence is that any exact Lagrangian in \(T^*S^{2k+1}\) whose Maslov class is zero is homeomorphic to \(S^{2k+1}\).NEWLINENEWLINEThe main result thus extends the monotone version of `Audin's conjecture'; i.e. that an embedded Lagrangian torus in \( \mathbb{C}^n\) has Maslov number 2. While this conjecture remains wide open at the time of writing, the monotone case had received considerable previous attention, having been verified for \(n \leq 24\) in [\textit{Y.-G. Oh}, ``Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings'', Int. Math. Res. Not. 1996, No. 7, 305--346 (1996; Zbl 0858.58017)], and then for all \(n\) in [\textit{K. Fukaya}, \textit{Y.-G. Oh}, \textit{H. Ohta}, and \textit{K. Ono}, Lagrangian intersection Floer theory: anomaly and obstruction. Part I. AMS/IP Studies in Advanced Mathematics 46,1. Providence, RI: American Mathematical Society (AMS); Somerville, MA: International Press (2009; Zbl 1181.53002)]. The latter reference also gives a fuller account of the history of the conjecture, and discusses extensions to the aspherical case.NEWLINENEWLINEThe main new tool in the paper under review is the introduction of a new version of Floer homology, which is defined by lifting the usual Floer complex to the universal cover \(\tilde{L}\) of \(L\). This `lifted' Floer homology, under the given conditions on \(L\), is the limit of a spectral sequence starting from the (Morse) homology of \(\tilde{L}\). The main results then follow by using properties of the homology of \(\tilde{L}\). As the author points out, this construction was hinted at in [\textit{J.-F. Barraud} and \textit{O. Cornea}, ``Quantization of the Serre spectral sequence'', J. Symplectic Geom. 5, No. 3, 249--280 (2007; Zbl 1153.55017)].
scientific article; zbMATH DE number 6043065