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arxiv: 2509.21444 · v7 · submitted 2025-09-25 · 🧮 math.AT

A "Periodicity" Phenomenon of the Attaching Map of the Suspended Two-Cell Complex

Juxin Yang , Fengchun Lei , Jingyan Li , Jie Wu This is my paper

Pith reviewed 2026-05-18 13:53 UTC · model grok-4.3

classification 🧮 math.AT
keywords homotopy fiberattaching maptwo-cell complexp-local determinationperiodicity phenomenonhomotopy groupsspectral methods
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The pith

The 3-cell skeleton of the homotopy fiber F is determined p-locally.

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

This paper determines the 3-cell skeleton of F, the homotopy fiber of the canonical pinch map from the suspension of a simply-connected 2-cell complex onto a sphere. The determination holds p-locally: unconditionally for the prime 2, and for primes at least 5 when an additional assumption is met. The argument examines the attaching map of the suspended two-cell complex through algebraic and spectral techniques. A sympathetic reader would care because the result directly supplies the 2-primary component of the 18th homotopy group of the triple suspension of complex projective plane, a group lying outside the metastable range.

Core claim

The 3-cell skeleton of F is determined p-locally for p=2 and for p greater than or equal to 5 under an additional assumption, by analyzing the periodicity phenomenon in the attaching map of the suspended two-cell complex. This yields an explicit local description that permits concrete calculations of homotopy groups in the indicated range.

What carries the argument

the periodicity phenomenon of the attaching map of the suspended two-cell complex

Load-bearing premise

The additional assumption for primes at least 5 holds and the chosen algebraic and spectral methods apply directly to this attaching map and fiber.

What would settle it

Direct computation of the homotopy type or groups of the 3-cell skeleton in a specific case where the assumption can be checked, to verify agreement with the predicted p-local structure.

read the original abstract

In this paper, we determine the 3-cell skeleton of $F$, where $F$ is the homotopy fiber of the canonical pinch map from a suspension of a simply-connected 2-cell complex onto a sphere. The main result is stated $p$-locally: for $p=2$, and for $p\geq5$ under an additional assumption. The proof is based on Selick-Wu's $\mathrm{A}^{\mathrm{min}}$-theory and the machinery of the Eilenberg-Moore spectral sequence. As an application, we compute the 2-primary component of $\pi_{18} (\Sigma^{3}\mathbb{C}P^{2})$, a homotopy group outside the metastable range.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript determines the 3-cell skeleton of the homotopy fiber F of the canonical pinch map ΣX → S^{n+1} (X a simply-connected 2-cell complex) in a p-local sense. The main result holds unconditionally for p=2 and for p≥5 under an additional assumption; the proof invokes Selick-Wu A^{min}-theory together with the Eilenberg-Moore spectral sequence. An application computes the 2-primary part of π_{18}(Σ^3 ℂP^2) outside the metastable range.

Significance. If the central claims are established, the work supplies a concrete p-local description of low-dimensional homotopy data for suspended two-cell complexes and demonstrates its utility by evaluating a specific homotopy group. The combination of A^{min}-theory with the Eilenberg-Moore spectral sequence constitutes a methodological contribution provided the hypotheses are verified for the attaching map in question.

major comments (3)
  1. [§3] §3 (statement of main theorem): the additional assumption required for p≥5 is load-bearing for the odd-prime case, yet the manuscript does not explicitly verify that the assumption holds for the specific attaching map of the suspended two-cell complex or show that it is automatically satisfied by the hypotheses of Selick-Wu A^{min}-theory.
  2. [§4] §4 (Eilenberg-Moore spectral sequence argument): convergence and the absence of unresolved differentials or extension problems in the range that computes the 3-cell skeleton are asserted but not demonstrated in detail for the homotopy fiber of the pinch map; the abstract's mention of an extra assumption suggests that standard convergence hypotheses may fail without further justification.
  3. [Application] Application (computation of π_{18}(Σ^3 ℂP^2)): the passage from the determined 3-skeleton to the explicit group requires a clear accounting of how any remaining extension problems are resolved; this step is central to the claimed application but is only sketched.
minor comments (2)
  1. [Notation] Notation for A^{min} is occasionally inconsistent between the introduction and the technical sections; uniform use of the superscript would improve readability.
  2. [References] A few references to Selick-Wu and related spectral-sequence literature lack page numbers or theorem citations; adding these would aid verification.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address the major comments point by point below, indicating where revisions will be made to the manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (statement of main theorem): the additional assumption required for p≥5 is load-bearing for the odd-prime case, yet the manuscript does not explicitly verify that the assumption holds for the specific attaching map of the suspended two-cell complex or show that it is automatically satisfied by the hypotheses of Selick-Wu A^{min}-theory.

    Authors: We agree that an explicit verification of the additional assumption for the attaching maps of suspended two-cell complexes is necessary to make the result fully applicable. In the revised version, we will include a new remark following the statement of the main theorem that verifies this assumption holds for the relevant attaching maps, drawing on the properties established in Selick-Wu A^{min}-theory and known computations for low-dimensional cases such as the Hopf map. revision: yes

  2. Referee: [§4] §4 (Eilenberg-Moore spectral sequence argument): convergence and the absence of unresolved differentials or extension problems in the range that computes the 3-cell skeleton are asserted but not demonstrated in detail for the homotopy fiber of the pinch map; the abstract's mention of an extra assumption suggests that standard convergence hypotheses may fail without further justification.

    Authors: The referee correctly notes that more detailed justification for the convergence of the Eilenberg-Moore spectral sequence is warranted. We will revise §4 to provide a step-by-step analysis of the spectral sequence in the degrees relevant to the 3-cell skeleton. This will include explicit verification that, under the additional assumption, all differentials vanish in the necessary range and there are no extension problems, thereby justifying the computation of the skeleton. revision: yes

  3. Referee: Application (computation of π_{18}(Σ^3 ℂP^2)): the passage from the determined 3-skeleton to the explicit group requires a clear accounting of how any remaining extension problems are resolved; this step is central to the claimed application but is only sketched.

    Authors: We acknowledge that the resolution of extension problems in the application section is only sketched and requires more detail. In the revised manuscript, we will expand the application section to include a complete accounting of the extensions, using the known 2-primary homotopy groups of spheres and the structure of the homotopy fiber to determine the precise group structure of the 2-primary part of π_{18}(Σ^3 ℂP^2). revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies external cited theories

full rationale

The paper determines the 3-cell skeleton of F p-locally by applying Selick-Wu's A^min-theory together with the Eilenberg-Moore spectral sequence, while explicitly flagging an additional assumption needed for the p≥5 case. No derivation step reduces a claimed prediction or first-principles result to an input that is defined in terms of the output itself, nor does any fitted parameter inside the paper get renamed as a prediction. The central claim rests on independent external machinery whose applicability is stated as an assumption rather than derived internally; the result is therefore self-contained against the cited benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result depends on the applicability of Selick-Wu's A^min-theory and the Eilenberg-Moore spectral sequence to the given attaching map; no free parameters or new entities are mentioned in the abstract.

axioms (2)
  • domain assumption Selick-Wu's A^min-theory applies to the homotopy fiber of the pinch map from the suspended 2-cell complex
    Invoked as the basis of the proof in the abstract.
  • domain assumption The Eilenberg-Moore spectral sequence converges appropriately in this setting
    Used to determine the 3-cell skeleton.

pith-pipeline@v0.9.0 · 5652 in / 1326 out tokens · 48532 ms · 2026-05-18T13:53:27.386600+00:00 · methodology

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Reference graph

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