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

Georgios Dimitroglou Rizell, Roman Golovko

TL;DR

The paper studies lower bounds for Reeb chords on a closed chord-generic Legendrian Λ⊂P×R that admits an exact Lagrangian filling L_Λ in the symplectisation, proving |Q(Λ)| ≥ stableMorse(L_Λ) under the spin and vanishing Maslov number hypotheses. It develops a simple-homotopy upgrade of Seidel’s isomorphism by identifying Floer homology with a twisted Morse complex after a small push-off and wrapping, and extends this to stabilized settings to derive bounds that depend on the homotopy type of L_Λ and on finite quotients of π_1(L_Λ) via Ono–Pajitnov-type invariants. These results improve previous Betti-number bounds and connect Reeb-chord counts to fundamental-group data, providing new examples with prescribed π_1 and showing that the new bounds can be strictly stronger than Seidel-type bounds. The methods combine twisted Floer theory, simple homotopy theory, and Ono–Pajitnov/ Sharko-type tools, and yield explicit constructions of exact Lagrangian fillings with various fundamental groups, including stabilised homology spheres and finite solvable groups. Overall, the work offers a robust framework for translating topological and group-theoretic data of Lagrangian fillings into sharp lower bounds on Reeb chords, with potential applications to Legendrian invariants and wrapped Floer theory.

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 $L_Λ$.

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

TL;DR

The paper studies lower bounds for Reeb chords on a closed chord-generic Legendrian Λ⊂P×R that admits an exact Lagrangian filling L_Λ in the symplectisation, proving |Q(Λ)| ≥ stableMorse(L_Λ) under the spin and vanishing Maslov number hypotheses. It develops a simple-homotopy upgrade of Seidel’s isomorphism by identifying Floer homology with a twisted Morse complex after a small push-off and wrapping, and extends this to stabilized settings to derive bounds that depend on the homotopy type of L_Λ and on finite quotients of π_1(L_Λ) via Ono–Pajitnov-type invariants. These results improve previous Betti-number bounds and connect Reeb-chord counts to fundamental-group data, providing new examples with prescribed π_1 and showing that the new bounds can be strictly stronger than Seidel-type bounds. The methods combine twisted Floer theory, simple homotopy theory, and Ono–Pajitnov/ Sharko-type tools, and yield explicit constructions of exact Lagrangian fillings with various fundamental groups, including stabilised homology spheres and finite solvable groups. Overall, the work offers a robust framework for translating topological and group-theoretic data of Lagrangian fillings into sharp lower bounds on Reeb chords, with potential applications to Legendrian invariants and wrapped Floer theory.

Abstract

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

Paper Structure

This paper contains 29 sections, 27 theorems, 51 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Results
  4. Previous results obtained using wrapped Floer homology
  5. Preliminaries
  6. Basics from symplectic and contact topology
  7. Floer homology with twisted coefficients
  8. The grading convention
  9. The stable Morse number
  10. Simple homotopy theory
  11. Group theoretic background
  12. The proof of Theorem \ref{['mainthmsmplehomeqnts']} and its consequences
  13. The main geometric construction
  14. Constructing a small push-off
  15. Wrapping
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $L_\Lambda \subset (\mathbb{R} \times P \times \mathbb{R},d(e^t\alpha))$ be an exact Lagrangian filling of an $n$-dimensional closed Legendrian submanifold $\Lambda \subset ( P \times \mathbb{R},\alpha)$ with fundamental group $\pi:=\pi_1(L_\Lambda)$ and Maslov number $\mu_{L_\Lambda} \in \mathb

Figures (2)

  • Figure 1: The union of $L_\Lambda$ together with its small Hamiltonian push-off $L_{\Lambda^\epsilon}=\phi^\epsilon_{e^t}(L_\Lambda)$. Note that the Reeb chords from the positive end of $L_{\Lambda^\epsilon}$ to the positive end of $L_\Lambda$ are in natural bijective correspondence with the Reeb chords on $\Lambda$.
  • Figure 2: After wrapping $L_\Lambda$ by applying the Hamiltonian flow $\Phi^s_{g\partial_z}$, the intersection points $\Phi^S_{g\partial_z}(L_\Lambda) \cap L_{\Lambda^\epsilon}$ are in natural one-to-one correspondence with the Reeb chords on $\Lambda$. The double point $p_c$ corresponds to the Reeb chord $c$ on $\Lambda$ which, in turn, corresponds to the Reeb chord $\widetilde{c}$ from $\Lambda^\epsilon$ to $\Lambda$.

Theorems & Definitions (44)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7: Seidel
  • Definition 2.1
  • Lemma 2.2: Lemma 4.1 OHRAFOELC
  • Definition 2.3
  • ...and 34 more