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

Abstract

Assume that we are given a closed chord-generic Legendrian submanifold $Λ\subset P \times \mathbb R$ of the contactisation of a Liouville manifold, where $Λ$ moreover admits an exact Lagrangian filling $L_Λ \subset \mathbb R \times P \times \mathbb R$ 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 $Λ$ is bounded from below by the stable Morse number of $L_Λ$. Given a general exact Lagrangian filling $L_Λ$, we show that the number of Reeb chords is bounded from below by a quantity depending on the homotopy type of $L_Λ$, 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 $Λ$ or $L_Λ$.