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If a Liouville domain is n-invertible, every Legendrian on its boundary has a Reeb chord of length at most the n-invertibility capacity.

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2026-07-12 13:51 UTC pith:32S3DOKJ

load-bearing objection Solid quantitative chord bound via a new capacity that settles Arnold for ST*T^n and submersion images of products of spheres.

arxiv 2606.16270 v2 pith:32S3DOKJ submitted 2026-06-15 math.SG

Lengths of Reeb chords and Viterbo restriction

Filip Bro\'ci\'c , Dylan Cant This is my paper
classification math.SG MSC 53D4253D1257R17
keywords Reeb chordsViterbo restrictionn-invertibilityLegendrianArnold chord conjecturesymplectic cohomologystring topologyrationality constant
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that the non-existence of short Reeb chords on a Legendrian forces an exact embedding of a stabilized disk cotangent bundle into the ambient Liouville domain. Viterbo restriction then transfers symplectic-cohomology structures, and when the domain is n-invertible the unit is forced to vanish after restriction, which is impossible for cotangent bundles. Consequently every compact Legendrian in the unit cotangent bundle of a manifold that admits a submersion from a product of spheres (for example the n-torus) admits a Reeb chord for every contact form. The same circle of ideas yields an upper bound on the rationality constants of aspherical non-exact Lagrangians inside the unit codisk bundle of the torus.

Core claim

Every Legendrian in the contact boundary of a Liouville domain Ω bounds a non-constant Reeb chord of period at most the n-invertibility capacity of Ω, whenever that capacity is finite. In particular the Arnold chord conjecture holds for the unit cotangent bundle of any manifold that is the base of a submersion from a product of spheres.

What carries the argument

n-invertibility capacity: the infimal action at which the unit of symplectic cohomology enters the smallest Δ-invariant ideal generated by the image of the top-degree PSS classes; non-vanishing of this capacity produces the contradiction after Viterbo restriction.

Load-bearing premise

The isomorphism that identifies string topology of the free loop space with symplectic cohomology of the cotangent bundle must preserve the products and BV operators that invert the top-degree classes.

What would settle it

An explicit compact Legendrian in the unit cotangent bundle of the n-torus that admits no Reeb chords for some contact form, or a computation showing that the unit never enters the ideal generated by the top PSS classes.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The paper introduces n-invertibility of a Liouville manifold W (Definition 1) and the associated capacity c_ni(Ω) (Definition 2), then proves that every Legendrian Λ ⊂ ∂Ω bounds a non-constant Reeb chord of period at most c_ni(Ω) whenever the capacity is finite (Theorem 3). The argument proceeds by constructing an exact embedding D_εT*Λ × D(a) when no short chords exist (Theorem 6), applying the Viterbo restriction map (Theorem 7) and a filtered Künneth isomorphism (Theorem 8) to force the unit to vanish in SH(T*Λ), and obtaining a contradiction with string topology. As applications, the authors show that T*(product of spheres) is n-invertible via explicit string-topology classes (Theorem 4), deduce the Arnold chord conjecture for ST*M whenever M admits a submersion from such a product (Theorem 10), and bound rationality constants of aspherical Lagrangians in unit disk cotangent bundles of tori (Theorem 5) via truncated Viterbo restriction (Theorem 11).

Significance. The work supplies a uniform, capacity-based upper bound on Reeb-chord lengths that recovers Zhou’s vanishing-SH results and extends them to aspherical bases such as T^n, where higher dilations and orientation tricks are unavailable. The geometric embedding of stabilized codisk bundles, the filtered Künneth map, and the truncated restriction for non-exact embeddings are carefully developed and of independent interest. The string-topology calculations that establish n-invertibility for products of spheres are topological and isolated from Floer theory, giving a clean, falsifiable criterion. Together these results settle the Arnold chord conjecture for a natural class of unit cotangent bundles and give the first rationality-constant bounds inside T*T^n.

minor comments (5)
  1. In Definition 1 and the subsequent persistence refinement, the notation for the ideal �_c and its colimit � is dense; a short clarifying sentence after Lemma 1.1 would help the reader track the filtration.
  2. Theorem 6 is stated for a Legendrian isotopy satisfying a strict inequality y*α > a dt; the subsequent reduction to the Reeb flow (Theorem 9) uses the non-strict version. A one-line remark on the limiting argument would remove any ambiguity.
  3. In §2.4.3 the homotopy between the classes P and D is continuous but not smooth; while the bordism-class model absorbs this, a parenthetical note that smoothing is possible would be reassuring.
  4. Appendix A.1.2 introduces the constants 7 and 3 without immediate motivation; a forward reference to the action-window estimates that follow would improve readability.
  5. A few typographical inconsistencies appear (e.g., “n-invertible domains always have dimension 2n” versus the later use of half-dimension convention; “�P^n” for RP^n). These are easily corrected.

Circularity Check

1 steps flagged

No significant circularity: capacity and geometric embedding are independent of the chord conclusion; self-citations to authors' prior Floer/string-topology framework transport topological calculations but do not force the result by definition.

specific steps
  1. self citation load bearing [§1.2.2 / Theorem 4 and surrounding text]
    "This proves that T*(S^{n1} imes⋯ imes S^{nk}) is n-invertible, due to the morphism of BV-algebras established in [Vit99, Vit96, AS06, Abo15, BC25]: H*(ΛM) o SH(T*M) sending [pt] to PSS(F) … Moreover, our argument also gives estimates on the n-invertibility capacity in terms of the lengths of the loops appearing in the classes A,B1,…,Bk, where the length functional is the one determined by the domain Ω⊂T*M considered in [BC25]."

    The topological calculation of the invertibility relations is independent, but the passage that makes T*(product of spheres) n-invertible (hence c_ni finite, hence chords exist by Thm 3) relies on the BV-algebra morphism whose filtered version and conventions are developed in the authors' own prior paper [BC25]. The citation is load-bearing for the concrete examples (including ST*T^n) yet does not render the chord statement true by definition; the geometric embedding + restriction steps remain independent.

full rationale

The derivation chain is: no short Reeb chords imply exact embedding of D_εT*Λ imes D(a) (Thm 6, pure geometry via coisotropic embedding + Gotay-type neighborhood), Viterbo restriction (Thm 7, standard + appendix) and filtered Künneth (Thm 8, constructed via split Hamiltonians and almost-complex structures) kill PSS(H^n) hence the unit in SH(T*Λ), contradicting the known non-vanishing of the unit under the Viterbo isomorphism. The capacity c_ni is defined independently via the persistence module of the Δ-invariant ideal generated by PSS(H^n); it is not fitted to chord data. For the product-of-spheres examples the invertibility relations A*Δ(B*[pt])=[S^n] (and products) are computed by direct bordism/string-topology manipulations of maps (completing manifolds, generalized sections, open-book classes) entirely inside H_*(ΛM); the only transport is the classical BV-algebra morphism H_*(ΛM) o SH(T*M) cited from Viterbo–Abbondandolo–Schwarz–Abouzaid plus the authors' prior technical paper [BC25]. That citation is load-bearing for the examples but parameter-free, does not assume the chord conclusion, and is mixed with external references; it does not make the existence statement true by construction. Truncated restriction and asphericity lemmas for rationality constants are likewise external or geometric. No self-definitional loop, no fitted-input-as-prediction, no uniqueness smuggled from the authors, and no renaming of a known empirical pattern. Score 1 reflects only the minor self-citation that is not definitional for the central claim.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 1 invented entities

The paper works entirely inside the standard package of Liouville domains, symplectic cohomology with PSS, BV and pair-of-pants product, and the Viterbo isomorphism with string topology. No free parameters are fitted. The only new entities are the n-invertibility notion and its capacity; both are defined by explicit algebraic operations already present in SH. Background results (Viterbo restriction, no-escape, Künneth for disks, string topology of spheres) are cited and used as black boxes.

axioms (4)
  • standard math Existence and basic properties of the persistence module SH^c(Ω), the PSS map, the BV operator Δ and the pair-of-pants product (standard Floer theory over a field of characteristic 2).
    Invoked throughout §1 and Appendix B; taken from Viterbo, Seidel, Abouzaid et al.
  • domain assumption Viterbo restriction (and its truncated version for non-exact embeddings) respects Δ, product and PSS, and maps free-homotopy summands correctly.
    Stated as Theorems 7 and 11; proved in Appendix A following Zhou and classical sources.
  • domain assumption The BV-algebra morphism H*(ΛM) → SH(T*M) of Abbondandolo–Schwarz–Abouzaid–Viterbo that sends [pt] to the PSS image of a cotangent fibre.
    Used in §1.2.2 and §2.5; the paper cites [Vit99, AS06, Abo15, BC25] and treats the morphism as given.
  • standard math No-escape lemma for Floer trajectories of the Viterbo Hamiltonian (AS10 Lemma 7.2).
    Invoked in §A.2 to identify the quotient complex with SH^b(K).
invented entities (1)
  • n-invertibility of a Liouville manifold (Definition 1) and the associated capacity c_ni(Ω) (Definition 2) independent evidence
    purpose: Quantifies how soon the top-degree PSS classes generate the unit under products and BV operators; supplies the upper bound on Reeb-chord lengths.
    Defined purely algebraically inside the already-existing persistence module SH^c; no new geometric object is postulated.

pith-pipeline@v1.1.0-grok45 · 44926 in / 2858 out tokens · 30732 ms · 2026-07-12T13:51:45.824496+00:00 · methodology

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read the original abstract

Let $\Lambda$ be a Legendrian in the contact boundary of a Liouville domain $\Omega$. We explain how the non-existence of Reeb chords with endpoints on $\Lambda$ of length up to $a$ enables one to embed $D_{\epsilon}T^{*}\Lambda\times D(a)$ into $\Omega$ in an exact way. As in earlier work of Zhengyi Zhou, we use the Viterbo restriction map to deduce a contradiction in certain cases. In particular, we show that if $M$ admits a submersion from a product of spheres (e.g., the $n$-torus), then all compact Legendrians $\Lambda\subset ST^{*}M$ admit a Reeb chord for every choice of contact form $ST^{*}M$. The obstruction we use in this case is based on the idea of inverting the degree-$n$ classes in cohomology, and is similar to the notion of string point invertibility introduced by Egor Shelukhin.

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  1. The strong Arnol'd chord conjecture for the boundary of a uniformly convex domain in $\mathbb{R}^{4}$

    math.SG 2026-06 unverdicted novelty 6.0

    Proves that any E3 Legendrian in the boundary of a Liouville domain bounds a chord of length at most liminf c_k(Ω)/k and applies this to establish the strong Arnol'd chord conjecture for uniformly convex domains in R^4.

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