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This paper proves that certain minimal Hamiltonian dynamics—pseudo-rotations and nontrivial finite-order actions—force quantum Steenrod powers to detect holomorphic spheres, making the ambient manifold geometrically uniruled, and derives ne

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load-bearing objection Strong conditional results: if the asserted adaptation of [BX26] holds, this resolves several open problems; the foundation is the main risk. the 4 major comments →

arxiv 2607.25960 v1 pith:BWZL7RVO submitted 2026-07-28 math.SG math.DS

Quantum Steenrod powers and Hamiltonian maps

Shaoyun Bai , Egor Shelukhin , Nicholas Wilkins , Guangbo Xu This is my paper
classification math.SG math.DS MSC 53D4053D4555S1037J1257S05
keywords quantum Steenrod powersHamiltonian pseudo-rotationsgeometric unirulednessFloer cohomologyConley conjecturesymplectically degenerate maximumequivariant pair-of-pants productperiodic points
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.

This paper is trying to establish an obstruction: on a closed symplectic manifold, some of the most economical Hamiltonian dynamics cannot exist unless the manifold is covered by holomorphic spheres. The precise claim is that if the manifold admits an F_p Hamiltonian pseudo-rotation, or a nontrivial Hamiltonian action by a finite group with p not dividing the group order, then the quantum Steenrod p-th power QSt_p(µ) of the counit differs from the classical Steenrod power St_p(µ); this "quantum deformation" is a nonzero Gromov–Witten type count and implies geometric uniruledness. The same equivariant Floer machinery yields dynamical corollaries: Hamiltonians with a periodic orbit in a non-torsion homology class and nonzero local Floer cohomology, or with a symplectically degenerate maximum, have infinitely many periodic points; and the generic Conley conjecture holds in new cases. If correct, it resolves a variant of McDuff–Salamon's Problem 24 and extends the Hofer–Zehnder program to arbitrary closed symplectic manifolds.

Core claim

In the paper's own terms, the central discovery is that the quantum Steenrod operation carries dynamical information that the ordinary Steenrod operation cannot see. For a closed symplectic manifold (M,ω), the p-th quantum Steenrod power QSt_p is a deformation of the classical total Steenrod power St_p by holomorphic-sphere counts; the paper proves that QSt_p(µ) ≠ St_p(µ) whenever M carries an F_p Hamiltonian pseudo-rotation or a nontrivial Hamiltonian action by a finite group with p not dividing the group order. Since a deformation means a nonzero contribution from a non-constant holomorphic sphere, and since the argument is uniform over points and almost complex structures, this yields geo

What carries the argument

The key object is the quantum Steenrod p-th power operation QSt_p: a power operation on quantum cohomology with coefficients in the Z/p-equivariant cohomology ring R_p = F_p[[t]]⊗E(θ), defined by counting holomorphic maps from a p-fold cover of a sphere with Z/p-equivariant perturbations and an additional marked point, so that its t,θ-expansion records corrections coming from holomorphic spheres. Around it sits the local equivariant pair-of-pants product, which relates the p-fold tensor power of local Floer cohomology of a Hamiltonian to the local equivariant Floer cohomology of the p-th iterate; the paper's divisibility lemma bounds the leading t-degree of this product by 2n(p−1) for p-admi

Load-bearing premise

The load-bearing premise is that the integral Floer-theoretic package built by the first and fourth authors adapts to cohomology, to continuation maps with varying Hamiltonians, and to the action-filtration framework with no more than cosmetic changes—something the paper states in Remark 2.2 as an implicit assumption rather than proving here; a second asserted extension is that the connecting-homomorphism argument for symplectically degenerate maxima carries over from Calabi–

What would settle it

Construct an F_p pseudo-rotation on a closed symplectic manifold that is not geometrically uniruled—for instance, a symplectically aspherical manifold, which carries no holomorphic spheres. The theorem says such a map cannot exist; producing one would refute Theorem 1.2. Alternatively, in an explicit monotone example such as the product X×X×CP² with X=Bl_6(CP²), compute QSt_2(u) directly for the Künneth product of point classes: the paper predicts QSt_2(u)=θ⁸u, and any different Gromov–Witten count would falsify Corollary 1.9.

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

If this is right

  • Any F_p Hamiltonian pseudo-rotation forces (M,ω) to be geometrically uniruled: for every compatible almost complex structure and every point, a nonconstant holomorphic sphere passes through that point.
  • A nontrivial finite-order Hamiltonian diffeomorphism forces the same uniruledness for every prime not dividing its order, and all its fixed points are contractible.
  • A Hamiltonian with a periodic orbit in a non-torsion homology class and nonvanishing local Floer cohomology has infinitely many simple periodic points; for sufficiently large primes these include simple periodic orbits in homology classes pγ.
  • A symplectically degenerate maximum implies infinitely many periodic points of the time-one map, with at least k²/log k growth up to period k.
  • Candidates for counterexamples to the generic Conley conjecture—finitely many hyperbolic periodic points—are ruled out for classes u of degree r>n satisfying a Steenrod-power condition, yielding new manifolds where the generic Conley conjecture holds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Editorial extension: the QSt_p(µ) ≠ St_p(µ) criterion is a purely quantum-cohomological obstruction to the existence of Hamiltonian pseudo-rotations; one could compute QSt_p on other monotone products to find further manifolds satisfying the generic Conley conjecture.
  • Editorial extension: the divisibility lemma suggests a broad local constraint—in local equivariant Floer cohomology of a p-admissible isolated fixed point, powers of t beyond 2n(p−1) must vanish in the image of the pants product—which may apply to other iteration problems beyond the ones treated here.
  • Editorial extension: the paper's chain of proof inherits any hidden failure of the asserted "cosmetic" adaptation of its companion Floer-theoretic package to cohomology and continuation maps; verifying that adaptation is the most direct way to test the truth of the Section 4 theorems.

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

4 major / 5 minor

Summary. The paper claims, for arbitrary closed symplectic manifolds, a series of results in Hamiltonian dynamics derived from integral Hamiltonian Floer theory: (i) existence of an F_p Hamiltonian pseudo-rotation or of nontrivial finite-order Hamiltonian action forces the quantum Steenrod p-th power QSt_p(µ) to differ from the classical Steenrod power St_p(µ), hence geometric uniruledness (Theorems 1.2, 1.3); (ii) a 1-periodic orbit in a non-torsion homology class with nontrivial local Floer cohomology, or a symplectically degenerate maximum, forces infinitely many simple periodic points (Theorems 1.16, 1.17); (iii) new cases of the generic Conley conjecture (Theorems 1.6, 1.11 and Corollaries 1.7, 1.9). The proofs use a homological-perturbation package, a new Künneth theorem for quantum Steenrod powers in the monotone case, and a divisibility lemma for the local equivariant pair-of-pants product.

Significance. If the foundational hypotheses are fully verified, these are substantial results. Theorems 1.2 and 1.3 address a variant of McDuff–Salamon Problem 24, Theorem 1.17 resolves a question of Ginzburg–Gürel, and Theorem 1.16 substantially generalizes prior non-contractible-orbit results. The Künneth property for QSt_p (Theorem 1.11) and the divisibility lemma (Lemma 4.1) are interesting in themselves and likely to be reused. The paper is also refreshingly explicit about which ingredients are taken as black boxes, and it makes a genuine effort to separate formal Floer-theoretic bookkeeping from the new geometric arguments. However, the central load-bearing input is a form of [BX26] that is not literally stated there; the manuscript explicitly labels the needed adaptation as an implicit cosmetic assumption. Until that adaptation, together with the asserted generalization of the symplectically degenerate maximum machinery, is proved or precisely located in the cited literature, the main theorems remain conditional.

major comments (4)
  1. [§2.1.2, Remark 2.2] Theorem 2.1 is stated as a free-module Floer cochain complex for arbitrary closed symplectic manifolds with action-filtration properties and continuation maps, but Remark 2.2 concedes that this is not the original statement of [BX26, Theorem A,B]: the original is Floer homology, and the needed variants with varying Hamiltonians, cohomological conventions, and action-filtration behavior are taken as an implicit assumption because the modifications are 'cosmetic'. This is load-bearing: every theorem in Section 4 inherits these properties through the filtered, local, and Tate constructions of Sections 2–3. Please provide either a proof of the adaptation or an exact theorem/lemma reference in [BX26] establishing this cohomological, Hamiltonian-varying, filtration-preserving form. Without this, the foundations of Theorems 1.2, 1.3, 1.6, 1.13, 1.16, and 1.17 are unverified.
  2. [§4.2, Lemma 4.2] Lemma 4.2 asserts that the connecting homomorphism in (4.2) is nontrivial for arbitrary closed symplectic manifolds, saying that the methods of [Gin10, Prop 4.7], [GG09a, Thm 1.18], and [Hei12, Thm 1.3] 'apply equally well' and 'extend, by applying Hein's decomposition'. No proof is supplied. This lemma is used directly in Theorem 1.17 and in Lemma 4.3 for pseudo-rotations, so the infinity of periodic points and the pseudo-rotation obstruction both rest on it. Please give a complete argument, or at least a precise statement in [Hei12] or [Gin10] that already covers arbitrary closed symplectic manifolds, since the cited results as stated in the text concern aspherical or Calabi–Yau settings.
  3. [§2.6.1, Theorem 2.22] Theorem 2.22, which gives invertibility of the local equivariant pants product and its nondegenerate form, is said to rely on a coproduct construction of [SZ21] 'which has not been fitted into the general framework of [BX26]'. This is not a merely cosmetic gap: Lemma 4.1 (the divisibility lemma) depends on Theorem 2.22, and Lemma 4.1 is used in the proof of Theorem 1.2 and in the pseudo-rotation argument. The paper needs either to integrate the [SZ21] coproduct into the [BX26] framework or to give a self-contained proof that the needed properties survive outside the weakly monotone setting.
  4. [§1.2.2, Remark 1.10; §3] Corollary 1.9 uses p=2 (quantum Steenrod squares), but the homological-perturbation results that carry the paper's applications, including Proposition 3.3, Corollary 3.5, Corollary 3.6, and Lemma 3.7, are stated only for odd primes. Remark 1.10 asserts that the odd-prime constructions 'work perfectly well' for p=2 with only the relation t=θ² changed, but no proof or even a precise list of the modified formulas is given. Since Corollary 1.9 is presented as a new case beyond [GG09b, Sug21], this extension is load-bearing for that corollary. Please provide the p=2 versions of the cited propositions or explicitly restrict Corollary 1.9 to a setting where the p=2 machinery is proved.
minor comments (5)
  1. [Abstract and headings] Typographical issues: 'G"urel' appears with raw TeX encoding in the abstract; 'Theorrem' appears in the headings of §4.5; 'T ate' appears in §2.5.1. These should be corrected.
  2. [§2.2.2] The Tate complex notation bC_{Z/p} and bH_{Z/p} is introduced, but in §2.5.1 and later the same objects are written dCF_{Z/p} and dHF_{Z/p}. The two notations should be reconciled or explicitly identified.
  3. [§2.4] The definition of QSt_p is given by a weighted sum over A∈H_2(M;Z). It may help to state explicitly that the Novikov variable q^A uses the symplectic area, matching equation (2.1)–(2.2), so that the convergence of the sum is clear to the reader.
  4. [§5.3] In the proof of Theorem 1.11, the statement says the left picture corresponds to κ_{Z/p}∘(QSt^{X1}_p⊗QSt^{X2}_p) and the right to QSt^{X1×X2}_p∘κ, but the sentence in the paragraph after Figure 3 reverses the order. The intended composition should be restated to avoid ambiguity.
  5. [§4.7, Claim D] The proof of Claim D is compressed to one sentence. Since this claim is used to identify the spectral carrier in the proof of Theorem 1.13, a short explanation of why μ_{Z/p,i}^{(p)}(t,θ) can only come from P^loc(μ_i^{(1)}) or P^big(μ) would improve readability.

Circularity Check

0 steps flagged

No definitional circularity; the central claims do not reduce to their inputs, though the proof rests heavily on an admitted, unproved adaptation of the same-authors' [BX26] package.

full rationale

I walked the derivation chain. The main theorems (1.2, 1.3, 1.6, 1.13, 1.16, 1.17) each proceed by contradiction or by combining the quantum Steenrod/equivariant-pants formalism with bar-length and local-Floer arguments; their conclusions (QSt_p(µ) ≠ St_p(µ), infinitely many periodic points) never appear as hypotheses. The dependency chain is acyclic: Proposition 3.1 → Corollary 3.2 → Lemma 4.3 → Theorem 1.2; Proposition 3.3/3.4 → Lemma 4.1; Lemma 4.2 is imported from Gin10/GG09a/Hei12 with the sentence 'The argument in [Hei12] applies equally well for an arbitrary closed symplectic manifold...' — a transferred result, not a self-referential input. No equation sets a target equal to a fitted parameter; no prediction is manufactured from a fitted constant. The only caveat worth flagging is Section 2.1.2, Remark 2.2: the authors explicitly state that the version of the [BX26] Floer package they need (cohomology, continuation with varying H, action filtration, integer coefficients) 'is not exactly the same as the original statement in loc. cit.' and that they 'take this as an implicit assumption as the modifications are cosmetic.' This is load-bearing and is a same-authors citation, so it is a verification gap and a reason not to give score 0; but it is not circular, because the target results are not contained in that assumption and the paper's own new arguments (divisibility lemma, Künneth propositions, spectral-carrier claims) carry independent content. Lemma 4.2's asserted extension to all closed symplectic manifolds is likewise an omitted proof, not a circular reduction. Overall: no significant circularity; score 2 reflects the self-citational foundational burden without a demonstrated equivalence of input and output.

Axiom & Free-Parameter Ledger

1 free parameters · 9 axioms · 0 invented entities

No numerical constants are fitted to data and no new physical entities (particles, forces, dimensions, conserved quantities) are postulated; the paper introduces new operations and constructions (quantum Steenrod powers at all primes, equivariant pair-of-pants products, Tate bar-length spectra, the divisibility lemma) rather than entities, so the invented-entities ledger is empty. The real credit the paper draws on is the [BX26] package (a self-cited foundational work by the first and fourth authors) plus multiple asserted transfers of prior arguments (Hein/Ginzburg, Sugimoto, GG10 normal forms). The only hand-chosen constant in the construction is the ε in the Morse function g on S^∞, which does not affect cohomology.

free parameters (1)
  • epsilon in the Z/p-invariant Morse function g on S^∞ (eq. 2.9) = unspecified small positive constant
    Hand-chosen constant in g(v) = Σ_l (l|v_l|² + ε·Re(v_l^p)) used to make the Morse model nondegenerate. The resulting R_p ring structure is independent of it, so it is not load-bearing for the main theorems.
axioms (9)
  • domain assumption [BX26] integral Hamiltonian Floer package (Z-coefficient Floer complexes; quantum Steenrod operations QSt_p; equivariant pants product; Tate isomorphisms) for arbitrary closed symplectic manifolds — quoted as Theorems 2.1, 2.10, 2.12, 2.17–2.19.
    All main results (Theorems 1.2, 1.3, 1.6, 1.13, 1.16, 1.17) are built on this package by the first and fourth authors; its correctness is not proved or independently checked in this paper.
  • ad hoc to paper Adaptations of [BX26] from Floer homology to Floer cohomology, and for continuation maps with varying Hamiltonians, are 'cosmetic' (Remark 2.2).
    The authors state: 'For all our discussions later on, similar comparisons need to be carried out ... We will take this as an implicit assumption as the modifications are cosmetic.' A load-bearing premise asserted, not demonstrated.
  • ad hoc to paper The nontrivial SDM connecting homomorphism (4.2) exists on arbitrary closed symplectic manifolds, by extending [Gin10, Prop 4.7], [GG09a, Thm 1.18], [Hei12, Thm 1.3].
    Lemma 4.2's proof says the argument in [Hei12] 'applies equally well' and the connecting map 'extends ... to the case of an arbitrary closed symplectic manifold'; the cited results are for Calabi–Yau/aspherical/rational settings.
  • domain assumption Sugimoto's filtered-Floer argument for non-torsion classes carries over verbatim (Theorem 1.16).
    Section 4.8: 'the argument of [Sug26, Proof of Theorem 2, Section 5] directly carries over'; the bar-length inequality (4.12) and the continuation diagram in the proof rely on [BX26, Thm 1.11] and standard continuation estimates.
  • domain assumption C²-generic Hamiltonian maps either have infinitely many periodic points or all-hyperbolic periodic points (Birkhoff–Lewis/Moser variant).
    Corollary 1.7 proof: 'A standard argument (see [GG09b, Sug21]) ... either all of its periodic points are hyperbolic or it has infinitely many periodic points by a variant of the Birkhoff-Lewis theorem [Mos77]'.
  • standard math Classical Steenrod operation degree bound: for |z|=k, the t^αθ^β component of St_p(z) vanishes when 2α+β > k(p−1).
    Used inside the Divisibility Lemma's claim (Section 4.1) to bound the totally degenerate factor; standard cohomology-operations theory.
  • domain assumption [GG10] normal-form reduction of a p-admissible isolated fixed point to a product of a totally degenerate and a nondegenerate factor.
    Proposition 2.16 and Lemma 4.1 reduce the local analysis via 'the argument of [GG10, Section 4.5]' — deformation with uniformly isolated fixed points; persistence of local (equivariant) Floer cohomology under such deformations is likewise taken from [GG10]/[SZ21].
  • ad hoc to paper Odd-prime constructions (Sections 2–3, Appendix B) adapt to p=2 with only the relation t=θ² changed.
    Remark 1.10: 'all the constructions for p>2 work perfectly well for p=2' — asserted for the F_2 calculations of Corollary 1.9, but the detailed constructions in the body are written for odd primes.
  • standard math Standard equivariant/Tate cohomology, Novikov field, quasi-Frobenius map, bar-length spectra formalisms (Sections 2.2–2.3).
    Background from [SW22, She22, UZ16, Sei15] used throughout; standard in the field.

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read the original abstract

We prove a series of new results in Hamiltonian dynamics on a general closed symplectic manifold $(M, \omega)$, including: 1. If $M$ admits a Hamiltonian diffeomorphism which is either a pseudo-rotation or has finite order, then $M$ is geometrically uniruled. This resolves a variant of Problem 24 in McDuff--Salamon's list, which predicts an obstruction to the existence of such special Hamiltonian diffeomorphisms in terms of genus zero numerical invariants. 2. If a Hamiltonian possesses a periodic orbit in a non-torsion homology class, then it has infinitely many simple periodic points. The same conclusion holds if the manifold is not geometrically uniruled and the diffeomorphism is minimal for rational Floer homology. These two general results complement the known cases of the Hofer--Zehnder conjecture. 3. If a Hamiltonian diffeomorphism possesses a symplectically degenerate maximum, then it has infinitely many simple periodic points. This resolves an open question which stems from the work of Ginzburg and G\"urel. We also establish new cases of the generic Conley conjecture: infinitely many periodic points for generic Hamiltonian diffeomorphisms. The proofs rely on a systematic application of the integral Hamiltonian Floer theory package developed by the first and fourth author, a K\"unneth isomorphism in equivariant Floer homology, and new quantitative analysis of quantum power maps, which is of independent interest.

Figures

Figures reproduced from arXiv: 2607.25960 by Egor Shelukhin, Guangbo Xu, Nicholas Wilkins, Shaoyun Bai.

Figure 1
Figure 1. Figure 1: Moduli space defining the equivariant Kunneth map for the tensor products. The coiled tree is a flow tree in S∞. which descends to the equivariant Kunneth map. One can also prove that on the cohomol￾ogy level, the equivariant Kunneth map is Rp-bilinear, where HMZ/p(f1), HMZ/p(f2), and HMZ/p(f1 × f2) are Rp-modules. We only summarize the results and omit the details. Proposition 5.1. The chain map (5.1) des… view at source ↗
Figure 2
Figure 2. Figure 2: Elements of Mw−,w+ (A; x1, . . . , xp, y), where w− = Z k i,ϵ and w+ = Z 0 j,δ [PITH_FULL_IMAGE:figures/full_fig_p057_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The cobordism leading to the Kunneth property of the quantum Steenrod operation. The coiled trees are flow trees in S∞ while the straight segments are flow lines in X1 or X2. The left picture corresponds to the composition κ Z/p ◦ (QStX1 p ⊗ QStX2 p ) while the right picture corresponds to the composition QStX1×X2 p ◦ κ. 5.3. Proof of Theorem 1.11. The proof is based on the TQFT/cobordism argument which is… view at source ↗
Figure 4
Figure 4. Figure 4: Kunneth property of the local equivariant Floer cohomology. The figure on the left corresponds to the composition κ loc Z/p◦(F Stloc p ⊗F Stloc p ) while the figure on the right corresponds to the composition F Stloc p ◦ κ loc . (3) From now on, we drop from the notations the dependence on H1 and H2 as all further constructions only depend on G1 and G2. To define the local equivariant complex for the produ… view at source ↗

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Reference graph

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