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Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings. (English) Zbl 0858.58017
It is well-known that for any compact Lagrangian embedding \(L \subset \mathbb{C}^n\), there exists a loop \(\gamma\) on \(L\) such that its Maslov index \(\mu_L (\gamma)\) satisfies \[ 3-n\leq\mu_L (\gamma) \leq n+1. \] When \(L\) is monotone, this can be improved by \[ 1\leq\mu_L (\gamma) \leq n+1. \] In the paper under review, the author obtains an optimal upper bound of the above inequalities. More exactly, let \(\Sigma_L\) be the minimal positive Maslov number of a compact monotone Lagrangian embedding \(L\subset \mathbb{C}^n\). Then we have: \[ 1 \leq \Sigma_L\leq n. \] The strategy of proof is to carefully analyse the Floer cohomology of our Lagrangian submanifold. This analysis leads him to some new and very deep results in the Floer cohomology. In conclusion, this is a nice work.

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
55N99 Homology and cohomology theories in algebraic topology
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