The stable Morse number as a lower bound for the number of Reeb chords

Assume that we are given a closed chord-generic Legendrian submanifold $\Lambda \subset P \times \mathbb{R}$ of the contactisation of a Liouville manifold, where $\Lambda$ moreover admits an exact Lagrangian filling $L_{\Lambda} \subset \mathbb{R} \times P \times \mathbb{R}$ inside the symplectisation. Under the further assumptions that this filling is spin and has vanishing Maslov class, we prove that the number of Reeb chords on $\Lambda$ is bounded from below by the stable Morse number of $L_{\Lambda}$. Given a general exact Lagrangian filling $L_{\Lambda}$, we show that the number of Reeb chords is bounded from below by a quantity depending on the homotopy type of $L_{\Lambda}$, following Ono–Pajitnov’s implementation in Floer homology of invariants due to Sharko. This improves previously known bounds in terms of the Betti numbers of either $\Lambda$ or $L_{\Lambda}$.

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Received 1 November 2015

Accepted 7 February 2018

Published 26 February 2019