Multiplicity of closed Reeb orbits on contact manifolds with periodic equivariant symplectic homology

Leonardo Macarini; Miguel Abreu

Multiplicity of closed Reeb orbits on contact manifolds with periodic equivariant symplectic homology

T0 review · 0 major / 4 minor · reviewed 2026-07-13 · grok-4.5

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Pith's one-line read A Floer-theoretic count gives a sharp lower bound on simple Reeb orbits, attained exactly when the form is lacunary.

desk verdict Solid, carefully proved sharp multiplicity bound for Reeb orbits under weak index assumptions on a large class of contact manifolds; the three-way characterization of r_M is the real advance. read the letter →

arxiv 2607.01219 v2 pith:NM552DBR submitted 2026-07-01 math.SG math.DS

Miguel Abreu , Leonardo Macarini This is my paper

classification math.SGmath.DS MSC 53D4037J1137J46

keywords ReeborbitsequivariantsymplectichomologymultiplicitylacunarycontactformConley-ZehnderindexprequantizationmeanEulercharacteristicBettinumbers

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

On a large class of closed contact manifolds whose positive equivariant symplectic homology is eventually periodic, the paper produces an explicit integer r_M built only from the contact Betti numbers and the mean Euler characteristic. For any non-degenerate contact form that avoids a few homologically unnecessary Conley-Zehnder indices, the number of simple closed Reeb orbits is at least r_M. Equality holds if and only if every orbit has Conley-Zehnder index of the same parity (the form is lacunary). Consequently, whenever a non-degenerate lacunary form exists, its orbit count equals r_M and is therefore a contact invariant completely determined by Floer theory. For prequantizations of symplectic orbifolds that admit a Hamiltonian circle action with isolated fixed points, the same number equals the total rank of the rational homology of the base, giving a purely topological reading of the invariant. The result unifies and sharpens earlier multiplicity theorems for spheres, toric contact manifolds, unit cosphere bundles of CROSSes, and Brieskorn spheres, and it motivates the conjecture that every contact form with finitely many closed orbits is necessarily non-degenerate and lacunary on a prequantization of this type.

What carries the argument

The integer r_M extracted from the eventually periodic positive equivariant symplectic homology HC_*(M): it measures the finite-dimensional defect between the initial sum of contact Betti numbers and the asymptotic average given by the mean Euler characteristic, and it becomes the exact orbit count of any non-degenerate lacunary form.

What would settle it

Exhibit a non-degenerate contact form on a manifold with periodic HC_* that has fewer than r_M simple orbits, or that attains r_M while possessing orbits of both even and odd Conley-Zehnder index.

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Extended reading notes

Core claim

Under weak index assumptions that exclude only good orbits of a single unnecessary degree, every non-degenerate contact form on a manifold with periodic positive equivariant symplectic homology has at least r_M simple closed Reeb orbits, where r_M is an explicit linear combination of contact Betti numbers and the positive mean Euler characteristic; equality holds if and only if the form is lacunary.

Load-bearing premise

The contact form must have no good closed orbit whose Conley-Zehnder index equals a prescribed integer of the wrong parity; the authors note this is probably technical but use it essentially for the Morse-inequality cancellations.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

This paper gives a sharp lower bound r_M on the number of simple closed Reeb orbits for non-degenerate contact forms on any closed contact manifold whose positive equivariant symplectic homology is periodic. The bound is extracted purely from the contact Betti numbers and the mean Euler characteristic; equality holds if and only if the form is lacunary. When a non-degenerate lacunary form exists, r_M is therefore a contact invariant equal to the number of its simple orbits. For the usual prequantizations (including toric ones and those of orbifolds with Hamiltonian circle actions) they also recover the topological count dim H_*(M/S^1;Q). That three-fold reading—homological, dynamical, topological—is new and cleanly unifies earlier results on spheres, CROSSes, and smooth prequantizations.

The technical work is standard but carefully assembled: Morse inequalities for the equivariant complex, resonance relations, and a refined index-recurrence statement (their Theorem 4). The main index hypothesis (no good orbits of index ±p of the “wrong” parity) is used essentially in the cancellations, but the authors show by explicit toric examples that it cannot be dropped for the wrong-parity bounds, while for the right-parity bounds of Theorem 1 it is automatic for any lacunary form. So the equality case is not artificially restricted. The proofs occupy Sections 5–8 and look algebraically self-contained once the index data are granted; I see no circularity or hidden dependence on the target multiplicity.

Soft spots are minor. The hypothesis is still technical (they say so themselves), and the general formula for non-lacunary homology is messier. The conjectures at the end are natural but open. None of this undercuts the main theorems.

This is for people working on Reeb dynamics, contact homology, or multiplicity problems in symplectic geometry. It deserves a serious referee; the arguments are checkable and the literature engagement is honest. I would cite the characterization of r_M and the toric/orbifold computations.

Referee Report

0 major / 4 minor

Summary. The paper studies closed contact manifolds with periodic positive equivariant symplectic homology HC_*(M). Under weak, homologically natural index assumptions on a non-degenerate contact form α (no good orbits of index ±p of opposite parity to the lacunary parity, non-vanishing mean indices, and index-admissibility under (NF)), it proves a sharp lower bound #P_α ≥ r_M, where r_M is an explicit combination of the contact Betti numbers and the positive mean Euler characteristic of M (Theorem 1, and the more general formulae of Theorems 5–6). Equality holds if and only if α is lacunary. Consequently, whenever a non-degenerate lacunary form exists, r_M equals the number of its simple closed Reeb orbits and is a contact invariant determined by Floer theory. The paper also obtains lower bounds for non-hyperbolic orbits (Theorem 3/7), computes r_M for prequantizations of monotone manifolds and orbifolds, good toric contact manifolds, Ustilovsky spheres, and manifolds with vanishing symplectic homology (Theorem 2), and shows that in the prequantization cases r_M = dim H_*(M/S^{1};ℚ). The proofs combine Morse inequalities for HC_*, resonance relations, and an index-recurrence theorem.

Significance. The work substantially generalizes and unifies earlier multiplicity results for spheres and prequantization bundles (Long–Zhu, Ginzburg–Gürel–Macarini, etc.) by replacing dynamical convexity or low-index exclusions with a weaker, homologically natural condition, and by treating the much larger class of manifolds with periodic HC_*. The invariant r_M is cleanly defined from Floer theory, admits dynamical and (in the prequantization case) topological characterizations, and is shown to be sharp precisely for lacunary forms. The computations cover essentially all currently known examples with finitely many Reeb orbits, and the conjectures in Section 9 are natural and well-motivated. The technical machinery (equivariant symplectic homology, Morse inequalities, resonance, index recurrence) is standard and carefully assembled; the necessity of the main index hypothesis is demonstrated by explicit toric counter-examples in §5.1.

minor comments (4)

The definition of r_M in Theorem 1 and the more general formulae (2.1) in Remark 2.9 / Theorems 5–6 are dense; a short table or display summarizing the four cases (lacunary/non-lacunary × parity of p) would improve readability without changing the mathematics.
In §6.2 the five classes of iterates (A)–(E) and the subsequent subclasses (B1±), (C1±) etc. are carefully tracked, but a brief schematic diagram or summary table of which classes contribute to the alternating sum would help the reader follow the cancellations leading to (6.17) and (6.20).
The dependence of HC_*(M) on the filling under condition (F) is correctly noted (and vanishes for lacunary forms); a single sentence in the introduction or §3.1 reminding the reader that all statements about r_M for lacunary forms are therefore filling-independent would make this point more prominent.
A few minor typos appear (e.g., “computer M” for “compute r_M”, occasional missing spaces around mathematical symbols). These are purely cosmetic.

Circularity Check

1 steps flagged · score 1.0 of 10

Homological r_M is defined independently of orbit counts; main multiplicity bound derived via Morse inequalities + index recurrence without feeding the target back into the inputs.

self citation load bearing [Theorem 4 (Index recurrence) and its proof]
"This theorem was proved in [29, Theorem 4.1] under the assumption that μ̂(Φ_i) > 0 for every i. We claim that the desired result can be reduced to this case."

The general index-recurrence statement used throughout §§6–7 is obtained by a short reduction to the authors’ earlier positive-mean-index result [29]. The reduction itself is elementary and non-circular, and [29] is an independent published paper; the step is therefore only a minor self-citation, not a load-bearing circularity that forces the multiplicity bound.

full rationale

r_M is introduced purely from the contact Betti numbers b_j and the mean Euler characteristic χ^{+}(M) of the (eventually) periodic positive equivariant symplectic homology HC_*(M) (Definition 1 and the explicit formulae in Theorem 1 / (2.1)). The inequality #P_α ≥ r_M and the equality case (iff α lacunary) are then proved in Theorems 5–6 by combining the Morse inequalities of Proposition 1, the resonance relation (3.1), and the index-recurrence Theorem 4; none of these steps redefine r_M in terms of the orbit count being bounded. When a non-degenerate lacunary form exists, Corollary 1 simply specialises the already-proved equality, yielding the dynamical characterisation. Self-citations ([3,6,8,29] etc.) supply either the combinatorial description of HC_* for toric examples or the positive-mean-index case of index recurrence; both are used as black-box inputs whose statements do not presuppose the multiplicity lower bound. The exclusion of good orbits of index ±p is an explicit hypothesis (not derived from the conclusion) and is shown necessary by counter-examples in §5.1. No fitted parameters, self-definitional loops, or uniqueness theorems imported solely from the authors appear in the load-bearing chain. The derivation is therefore self-contained against its own Floer-theoretic inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The paper works entirely within the standard axiomatic framework of contact geometry and Floer theory. The only non-standard inputs are the periodicity assumption on HC_* (Definition 1) and the technical exclusion of orbits of index ±p; both are stated explicitly and their necessity is discussed.

assumptions (4)

domain assumption Positive equivariant symplectic homology HC_*(M) is well-defined with integer grading under either a strong symplectic filling with c_1(TW)=0 and H^1(M;Q)→H^1(W;Q) trivial, or c_1(ξ)=0, H^1(M;Q)=0 and existence of an index-admissible form (conditions (F) or (NF)).
Standard foundational hypotheses guaranteeing that the Conley–Zehnder index is independent of trivialization choices; invoked throughout Sections 2–3.
domain assumption HC_*(M) is eventually periodic with even period Δ after degree P (Definition 1).
The central hypothesis that defines the class of manifolds under study; used to guarantee that the mean Euler characteristic exists and that r_M is independent of the choice of P and Δ.
domain assumption Every closed orbit of the non-degenerate form α has non-vanishing mean index.
Ensures that Morse-type numbers are finite and that the resonance relation applies; automatically satisfied by lacunary forms when all Betti numbers are finite.
standard math Index-recurrence theorem (Theorem 4) for a finite collection of strongly non-degenerate paths in the universal cover of Sp(2n).
Generalization of the common-index-jump theorem of Long–Zhu and Duan–Liu–Long–Wang; proved in Section 4 by reduction to the positive-mean-index case.

invented entities (1)

r_M (the multiplicity invariant extracted from contact Betti numbers) independent evidence
purpose: Provides a single integer that is simultaneously a lower bound for the number of simple Reeb orbits, a dynamical count for lacunary forms, and (for prequantizations) a topological Betti number.
Defined purely in terms of the already-existing positive equivariant symplectic homology; no new geometric object is postulated.

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Pith. "Pith review of Multiplicity of closed Reeb orbits on contact manifolds with periodic equivariant symplectic homology." pith.science (2026). https://pith.science/paper/NM552DBR

@misc{pith2026260701219,
  author       = {Pith},
  title        = {Pith review of: Multiplicity of closed Reeb orbits on contact manifolds with periodic equivariant symplectic homology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NM552DBR}},
  note         = {Machine review of arXiv:2607.01219}
}

abstract

We consider closed contact manifolds $(M,\xi)$ with periodic positive equivariant symplectic homology. This is a very large class of contact manifolds and, to the best of our knowledge, includes all currently known examples admitting Reeb flows with finitely many closed orbits for which this homology is well defined. Under weak and homologically natural index assumptions on a non-degenerate contact form $\alpha$ on $M$, we establish a sharp lower bound $r_M$ for the number of simple closed Reeb orbits of $\alpha$. The quantity $r_M$ is completely defined in terms of the positive equivariant symplectic homology of $M$. Moreover, we show that this bound is attained if and only if $\alpha$ is lacunary, i.e., the Conley-Zehnder indices of all closed orbits have the same parity. Consequently, the invariant $r_M$ admits a clean dynamical characterization: whenever $M$ admits a non-degenerate lacunary contact form, $r_M$ equals the number of its simple closed Reeb orbits and is therefore independent of the choice of such a form. In particular, in the lacunary case the number of such orbits is a contact invariant completely determined by Floer theory. We compute $r_M$ for a broad class of examples, including several prequantizations of symplectic orbifolds, and show that in this case $r_M=\dim H_*(M/S^1;\mathbb{Q})$, thereby giving a topological characterization of the invariant. Motivated by these results, we conjecture that any contact form with finitely many closed Reeb orbits is necessarily non-degenerate and lacunary, and that the underlying contact manifold is a prequantization of this type.

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[1] [1]

Abbondandolo, M

A. Abbondandolo, M. Schwarz,On the Floer homology of cotangent bundles.Comm. Pure Appl. Math. 59(2006), 254–316

2006

[2] [2]

Abreu, J

M. Abreu, J. Gutt, J. Kang, L. Macarini,Two closed orbits for non-degenerate Reeb flows.Math. Proc. Cambridge Philos. Soc.170(2021), no. 3, 625–660

2021

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Abreu, L

M. Abreu, L. Macarini,Contact homology of good toric contact manifolds.Compositio Mathematica148 (2012), 304–334

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M. Abreu, L. Macarini,Multiplicity of periodic orbits for dynamically convex contact forms.Journal of Fixed Point Theory and Applications19(2017), 175–204

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Abreu, L

M. Abreu, L. Macarini,On the mean Euler characteristic of Gorenstein toric contact manifolds.Int. Math. Res. Not. IMRN 2020, no. 14, 4465–4495

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Abreu, L

M. Abreu, L. Macarini,Periodic orbits of non-degenerate lacunary contact forms on prequantization bundles.Selecta Math. (N.S.)32(2026), no. 3, Paper No. 59. MULTIPLICITY OF CLOSED REEB ORBITS 71

2026

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M. Abreu, L. Macarini, M. Moreira,On contact invariants of non-simply connected Gorenstein toric contact manifolds.Mathematical Research Letters29(2022), 1–42

2022

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Abreu, L

M. Abreu, L. Macarini, M. Moreira,Contact invariants ofQ-Gorenstein toric contact manifolds, the Ehrhart polynomial and Chen-Ruan cohomology.Advances in Mathematics429(2023), Paper No. 109180

2023