The stable Morse number as a lower bound for the number of Reeb chords
Abstract
Assume that we are given a closed chordgeneric 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 OnoPajitnov'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 $L_{\Lambda}$.
 Publication:

arXiv eprints
 Pub Date:
 October 2015
 arXiv:
 arXiv:1510.08838
 Bibcode:
 2015arXiv151008838D
 Keywords:

 Mathematics  Symplectic Geometry;
 Mathematics  Geometric Topology;
 53D12 (Primary);
 53D42 (Secondary)
 EPrint:
 37 pages, 2 figures, final version