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The chord conjecture for conormal bundles

Filip Broćić, Dylan Cant, Egor Shelukhin

TL;DR

The paper proves Arnol'd's chord conjecture for conormal Legendrians in $S^{*}M$ by establishing a twisted Viterbo-type isomorphism between the positive wrapped Floer cohomology $\\mathrm{HW}_{+}(\\nu^{*}N;\\mathscr{L})$ and the relative path-space homology $H_{*}(\\mathscr{P},N;\\mathscr{L})$ with local coefficients. A twisted relative Hurewicz theorem ensures nontrivial relative homology for an appropriate local system $\\mathscr{L}$, which, via the isomorphism, implies the existence of Reeb chords for the ideal boundary of the conormal, hence proving Arnol'd's chord conjecture in this setting. The approach combines Morse theory for the energy functional with local coefficients, a robust theory of positive wrapped Floer cohomology for conormal Lagrangians, and careful orientation and compactness analyses to relate Morse and Floer theories. This framework extends chord-conjecture results to conormal bundles of submanifolds and provides a versatile method for producing Reeb chords via homotopical and homological data, potentially applicable to broader nonlocal boundary problems in symplectic topology.

Abstract

We prove Arnol'd's chord conjecture for all Legendrian submanifolds of cosphere bundles of closed manifolds isotopic to conormal bundles of closed submanifolds. Our method of proof involves an isomorphism between wrapped Floer cohomology and the homology of a path space with coefficients in a local system and a twisted version of the Hurewicz theorem.

The chord conjecture for conormal bundles

TL;DR

The paper proves Arnol'd's chord conjecture for conormal Legendrians in by establishing a twisted Viterbo-type isomorphism between the positive wrapped Floer cohomology and the relative path-space homology with local coefficients. A twisted relative Hurewicz theorem ensures nontrivial relative homology for an appropriate local system , which, via the isomorphism, implies the existence of Reeb chords for the ideal boundary of the conormal, hence proving Arnol'd's chord conjecture in this setting. The approach combines Morse theory for the energy functional with local coefficients, a robust theory of positive wrapped Floer cohomology for conormal Lagrangians, and careful orientation and compactness analyses to relate Morse and Floer theories. This framework extends chord-conjecture results to conormal bundles of submanifolds and provides a versatile method for producing Reeb chords via homotopical and homological data, potentially applicable to broader nonlocal boundary problems in symplectic topology.

Abstract

We prove Arnol'd's chord conjecture for all Legendrian submanifolds of cosphere bundles of closed manifolds isotopic to conormal bundles of closed submanifolds. Our method of proof involves an isomorphism between wrapped Floer cohomology and the homology of a path space with coefficients in a local system and a twisted version of the Hurewicz theorem.
Paper Structure (81 sections, 26 theorems, 128 equations, 22 figures)

This paper contains 81 sections, 26 theorems, 128 equations, 22 figures.

Table of Contents

  1. Introduction
  2. Local systems of coefficients
  3. Structure of the paper
  4. Proofs
  5. Morse homology for the energy functional
  6. Finite dimensional approximation
  7. Admissible pseudogradients
  8. Morse differential twisted by local coefficients
  9. Continuation lines
  10. Definition of the full Morse homology
  11. Comparison with relative singular homology
  12. Compatibility with continuation lines
  13. Induced map on the colimit
  14. On the choice of local system and the twisted Hurewicz theorem
  15. Free local systems
  16. ...and 66 more sections

Key Result

Theorem 1.1

Let $N \subset M$ be a closed submanifold of a closed manifold with positive codimension. Arnol'd's chord conjecture holds for $\Lambda_N$ in $S^*M.$

Figures (22)

  • Figure 1: The unstable sphere $S(x)$ contains the point $z$ which lies on the flow line joining $x$ to $y$.
  • Figure 2: Continuation line $\xi(s)$ from $(\mathscr{P}^{\ell}_{K},V)$ to $(\mathscr{P}^{\ell'}_{dK},V')$.
  • Figure 3: Attaching a 1-handle to cross the critical point $q$. The solid line is pre-surgery, the dashed line is post-surgery.
  • Figure 4: The one-dimensional component $C$ of the moduli space of index difference two chords.
  • Figure 5: Gluing linearized operators when proving $d^2=0$; the two ends of the moduli space correspond to setting $i=1,2$.
  • ...and 17 more figures

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • ...and 18 more