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 ⊂ P × R of the contactisation of a Liouville manifold, where Λ moreover admits an exact Lagrangian filling L Λ ⊂ R × P × 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 .

Simple homotopy type of the Hamiltonian Floer complex

“…For M with boundary, the argument of Damian can be adopted slightly as in Proposition 2.9 in [5] to conclude similarly. Here the stable Morse functions h are required to be of the form h = Q + f outside a compact set in the interior, where f is a function on M having ∂ M contained in a regular level set.…”

“…Simple homotopy equivalence is an improvement compared to Floer's result of homotopy equivalence [8]. Simple homotopy and the related notion of torsion has been studied for Floer complexes for instance in [16] [12] [1] [15] [5]. The complex CF * (H) also appears in [14].…”