Secondary Ext of the Fiber of $Sq^n$ and the Secondary Adams Spectral Sequence

Dang Vo Phuc

arxiv: 2605.18906 · v1 · pith:DHNIHUIBnew · submitted 2026-05-17 · 🧮 math.AT

Secondary Ext of the Fiber of Sq^n and the Secondary Adams Spectral Sequence

Dang Vo Phuc This is my paper

Pith reviewed 2026-05-20 12:54 UTC · model grok-4.3

classification 🧮 math.AT

keywords secondary Steenrod algebrasecondary Ext groupsfiber of Sq^nsecondary Adams spectral sequenceBaues-Jibladze comparison theoremBruner's E3 calculationtrack-fiber realization

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The pith

The secondary Ext groups of the fibers of Sq^n over the secondary Steenrod algebra are explicit direct sums of free modules and suspensions thereof.

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

The paper computes the secondary Ext groups Ext_B of the secondary cohomology of three specific fibers associated to the Steenrod square Sq^n at the prime 2. For the fiber F_n it finds a two-term direct sum of copies of F2 in bidegrees (0,0) and (1,n). For F_nZ the result adds a polynomial factor F2[h0] on the first summand. For the infinite product fiber F the computation includes both the polynomial factor and an infinite direct sum over odd degrees that are not powers of 2. These isomorphisms are obtained by combining Bruner's earlier E3-page calculation with the Baues-Jibladze comparison theorem between secondary and ordinary Ext; an alternative route via explicit secondary mapping fibers is available once a track-fiber realization hypothesis is granted.

Core claim

At the prime 2, the secondary Steenrod algebra B yields Ext_B^{*,*}(H_B^* F_n, F2) ≅ F2 ⊕ Σ^{1,n} F2, Ext_B^{*,*}(H_B^* F_{nZ}, F2) ≅ F2[h0] ⊕ Σ^{1,n} F2, and Ext_B^{*,*}(H_B^* F, F2) ≅ F2[h0] ⊕ ⊕_{j>0} Σ^{1,2^j} F2 ⊕ ⊕_{i>0, i≠2^k} Σ^{0,2i-1} F2, where the last sum runs over positive integers i that are not powers of 2 and h0 has bidegree (1,1). The group-level results follow directly from Bruner's E3-calculation together with the Baues-Jibladze comparison theorem, which also supplies explicit primary descriptions of the kernels and cokernels of the maps D_n, D_{nZ} and D_F. Under the track-fiber realization hypothesis the same groups arise from displayed secondary mapping fibers, and the d

What carries the argument

Baues-Jibladze comparison theorem between secondary and ordinary Ext groups, applied to Bruner's E3-page for the fibers of Sq^n

If this is right

The E2-page of the secondary Adams spectral sequence for each of these fibers is now known explicitly.
The first differential in secondary homological algebra is the primary shadow of the ordinary Adams d2 via the Bruner-Rognes Yoneda description.
Kernels and cokernels of the maps D_n, D_nZ and D_F admit explicit primary descriptions.
Higher pages of the secondary spectral sequence for the homotopy of these fibers become accessible once the Ext groups are inserted.

Where Pith is reading between the lines

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

The same comparison technique may apply to other secondary cohomology objects whose ordinary Ext is already known from classical Adams spectral sequence calculations.
If the track-fiber hypothesis can be verified for a larger class of maps, secondary Adams spectral sequences could be computed routinely for many more spaces.
The pattern of summands indexed by non-powers of 2 suggests a connection to the structure of the image of J or to the 2-primary Kervaire invariant problem.

Load-bearing premise

The track-fiber realization hypothesis must hold in order to obtain the groups from explicit secondary mapping fibers rather than from the primary comparison theorem alone.

What would settle it

An independent calculation of the secondary cohomology rings of F_n, F_nZ and F followed by a direct computation of their Ext groups over the secondary Steenrod algebra B, checking whether the resulting modules match the stated direct sums.

read the original abstract

At the prime $2$, let $\mathcal{B}$ denote the secondary Steenrod algebra in the sense of Baues, Baues--Jibladze, Nassau, and Baues--Frankland. We determine the secondary Ext groups of the secondary cohomology objects of the three fibers considered by Bruner in his work on the fiber of $\mathrm{Sq}^n$. For $$F_n = \mathrm{fib}(H \xrightarrow{\mathrm{Sq}^n} \Sigma^n H),\,\, \text{and}\,\, F_{n\mathbb{Z}} = \mathrm{fib}(H\mathbb{Z} \xrightarrow{\mathrm{Sq}^n} \Sigma^n H),$$ where $n>0$ in the first case and $n \ge 2$ in the second, and for $$F = \mathrm{fib}(H\mathbb{Z} \to \prod_{i>0} \Sigma^{2i} H),$$ where the $i$-th component is represented after mod-$2$ reduction by $\mathrm{Sq}^{2i}$, the groups are $$\mathrm{Ext}_{\mathcal{B}}^{*,*}(H^*_{\mathcal{B}} F_n, \mathbb{F}_2) \cong \mathbb{F}_2 \oplus \Sigma^{1,n}\mathbb{F}_2,$$ $$\mathrm{Ext}_{\mathcal{B}}^{*,*}(H^*_{\mathcal{B}} F_{n\mathbb{Z}}, \mathbb{F}_2) \cong \mathbb{F}_2[h_0] \oplus \Sigma^{1,n}\mathbb{F}_2,$$ and $$\mathrm{Ext}_{\mathcal{B}}^{*,*}(H^*_{\mathcal{B}} F, \mathbb{F}_2) \cong \mathbb{F}_2[h_0] \oplus \bigoplus_{j>0} \Sigma^{1,2^j}\mathbb{F}_2 \oplus \bigoplus_{i>0, i \neq 2^k} \Sigma^{0,2i-1}\mathbb{F}_2$$ (where the last direct sum runs over $i$ not a power of $2$). Here $h_0$ has bidegree $(1,1)$. The group-level calculation follows from Bruner's $E_3$-calculation and the Baues--Jibladze comparison theorem, and the proof gives an explicit primary calculation of the kernels and cokernels of the maps $D_n$, $D_{n\mathbb{Z}}$, and $D_F$. Separately, under an explicit track-fiber realization hypothesis, the same groups are computed from displayed secondary mapping fibers. Under the same hypothesis, the Bruner--Rognes Yoneda description of the ordinary Adams $d_2$ is the primary shadow of the first differential in secondary homological algebra.

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.

Desk Editor's Note private letter to a colleague

This paper gives explicit secondary Ext isomorphisms for Bruner's Sq^n fibers via Bruner E3 plus Baues-Jibladze, but the secondary mapping-fiber route is conditional on a track-fiber hypothesis.

read the letter

The main point is that the author works out the secondary Ext groups over the secondary Steenrod algebra for the cohomology of three specific fibers: fib(Sq^n) from H, the version from HZ, and the big fiber from the product of all Sq^{2i}. The results are clean direct sums: just F2 plus a shifted F2 for the first, F2[h0] plus the shift for the second, and F2[h0] plus shifts at powers of 2 and odd degrees for the third. These descriptions are new in this explicit form and not already written down in Bruner or Baues-Jibladze. The paper also supplies the primary kernel and cokernel calculations for the maps D_n, D_nZ, and D_F that feed into the comparison theorem, which is the concrete algebraic work that actually produces the isomorphisms. That part looks solid and directly useful for anyone feeding data into a secondary Adams spectral sequence. The connection to the Bruner-Rognes Yoneda description of the ordinary d2 as the primary shadow of the first secondary differential is a nice extra observation. The soft spot is the track-fiber realization hypothesis required for the secondary mapping-fiber computation of the same groups. The abstract presents this as a separate route, so if the hypothesis does not hold for these particular fibers, only the primary calculation via Bruner's E3 and the comparison theorem carries the load. Without independent verification that the secondary module structures on H_B^* F_* match the claimed Ext exactly, that part stays conditional. The rest of the argument does not appear to introduce circularity or free parameters. This is specialized work aimed at people already computing stable homotopy groups at p=2 with secondary operations or Adams spectral sequences. A reader who needs concrete algebraic input for differentials will find the explicit groups helpful. It deserves a serious referee who can check the primary kernel calculations and clarify how much the hypothesis can be relaxed or confirmed.

Referee Report

2 major / 3 minor

Summary. The paper computes the secondary Ext groups Ext_B^{*,*}(H_B^* F, F_2) for three fibers F_n = fib(H → Σ^n H via Sq^n), F_{nZ} = fib(HZ → Σ^n H via Sq^n), and F = fib(HZ → ∏ Σ^{2i} H via Sq^{2i}) in the secondary Steenrod algebra B at p=2. It claims the isomorphisms Ext_B^{*,*}(H_B^* F_n, F_2) ≅ F_2 ⊕ Σ^{1,n} F_2, Ext_B^{*,*}(H_B^* F_{nZ}, F_2) ≅ F_2[h_0] ⊕ Σ^{1,n} F_2, and Ext_B^{*,*}(H_B^* F, F_2) ≅ F_2[h_0] ⊕ ⊕_{j>0} Σ^{1,2^j} F_2 ⊕ ⊕_{i>0, i≠2^k} Σ^{0,2i-1} F_2. These follow from Bruner's E_3-calculation and the Baues-Jibladze comparison theorem with explicit primary kernel/cokernel computations for the maps D_n, D_{nZ}, D_F. Separately, under a track-fiber realization hypothesis, the same groups arise from displayed secondary mapping fibers, and this relates the Bruner-Rognes Yoneda description of the ordinary Adams d_2 to the first differential in secondary homological algebra.

Significance. If the results hold, this provides explicit computations of secondary Ext groups for objects central to the secondary Adams spectral sequence at p=2. By combining Bruner's E_3-calculation with the Baues-Jibladze comparison theorem and supplying explicit primary kernel/cokernel calculations, the work offers a concrete and verifiable route to these groups. The alternative secondary-fiber computation under the stated hypothesis supplies an independent perspective that may clarify differentials in the Adams spectral sequence.

major comments (2)

Abstract, final paragraph: the group-level isomorphisms are asserted to follow from Bruner's E_3-calculation and the Baues-Jibladze comparison theorem together with explicit primary kernel/cokernel calculations for D_n, D_{nZ}, D_F; the manuscript must display these calculations in full so that the claimed direct-sum decompositions can be verified directly from the primary data.
Abstract, final paragraph: the same Ext groups are obtained from displayed secondary mapping fibers only under the explicit track-fiber realization hypothesis; the paper should state whether this hypothesis has been verified for the concrete fibers fib(Sq^n) and fib(Sq^{2i}), because failure of the hypothesis would render the secondary route unavailable and place the entire weight of the isomorphisms on the comparison-theorem route.

minor comments (3)

The secondary Steenrod algebra B and the secondary cohomology functor H_B^* should be recalled with a short paragraph in the introduction for readers who may not have the Baues-Jibladze-Nassau framework at hand.
Bidegrees are indicated for h_0 ((1,1)) but should be stated uniformly for all summands in the three displayed isomorphisms.
The relation between the secondary first differential and the Bruner-Rognes Yoneda description of the Adams d_2 is stated only in the abstract; a brief sentence in the introduction would clarify the broader context.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which we address point by point below. The main results rest on the Baues–Jibladze comparison theorem together with explicit primary computations, and we will revise the manuscript to improve clarity and verifiability as requested.

read point-by-point responses

Referee: Abstract, final paragraph: the group-level isomorphisms are asserted to follow from Bruner's E_3-calculation and the Baues-Jibladze comparison theorem together with explicit primary kernel/cokernel calculations for D_n, D_{nZ}, D_F; the manuscript must display these calculations in full so that the claimed direct-sum decompositions can be verified directly from the primary data.

Authors: We agree that the explicit kernel and cokernel computations for the maps D_n, D_{nZ}, and D_F should be displayed in full to permit direct verification. The revised manuscript will include a new subsection (or appendix) that records these primary calculations in detail, including the relevant exact sequences and the resulting direct-sum decompositions of the secondary Ext groups. revision: yes
Referee: Abstract, final paragraph: the same Ext groups are obtained from displayed secondary mapping fibers only under the explicit track-fiber realization hypothesis; the paper should state whether this hypothesis has been verified for the concrete fibers fib(Sq^n) and fib(Sq^{2i}), because failure of the hypothesis would render the secondary route unavailable and place the entire weight of the isomorphisms on the comparison-theorem route.

Authors: The manuscript already presents the secondary mapping-fiber computation as conditional on the track-fiber realization hypothesis. In the revision we will add an explicit remark clarifying that this hypothesis has not been verified in the present work for the specific fibers fib(Sq^n) and fib(Sq^{2i}). Consequently the unconditional proof of the stated isomorphisms proceeds via Bruner's E_3-calculation and the Baues–Jibladze comparison theorem; the secondary route is offered only as an independent perspective under the stated hypothesis. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations rely on external Bruner E3-calculation and Baues-Jibladze comparison theorem

full rationale

The paper states that the claimed Ext isomorphisms follow directly from Bruner's prior E3-calculation combined with the Baues-Jibladze comparison theorem, together with explicit primary kernel/cokernel computations for the maps D_n, D_nZ, and D_F. The secondary mapping-fiber route is presented separately and is explicitly conditional on an additional track-fiber realization hypothesis. No step in the provided derivation chain reduces by construction to a self-definition, a fitted parameter renamed as a prediction, or a load-bearing self-citation whose content is unverified outside the present work. The cited results are independent external theorems, so the central claims remain non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The computations rest on the existence and basic properties of the secondary Steenrod algebra as defined by Baues et al., the validity of the Baues-Jibladze comparison theorem, and Bruner's prior E3-page calculation; no free parameters or new invented entities are introduced.

axioms (2)

domain assumption The secondary Steenrod algebra B exists and satisfies the comparison theorem of Baues-Jibladze.
Invoked to equate the secondary Ext computation with the primary kernel/cokernel calculation.
domain assumption Bruner's E3-calculation for the fibers is correct and can be lifted to the secondary setting.
Used as the starting point for determining the secondary Ext groups.

pith-pipeline@v0.9.0 · 6062 in / 1570 out tokens · 27226 ms · 2026-05-20T12:54:45.344339+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Hypothesis 3.4 (Track-fiber realization) ... secondary mapping fiber Fib_B(f♯)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages · 1 internal anchor

[1]

J. F. Adams, On the structure and applications of the Steenrod algebra,Commentarii Mathematici Helvetici 32(1958), 180–214, DOI:https://doi.org/10.1007/BF02564578

work page doi:10.1007/bf02564578 1958
[2]

247, Birkh¨ auser Basel, 2006, DOI:https://doi.org/10.1007/3-7643-7449-7

H.-Joachim Baues,The Algebra of Secondary Cohomology Operations, Progress in Mathematics, vol. 247, Birkh¨ auser Basel, 2006, DOI:https://doi.org/10.1007/3-7643-7449-7

work page doi:10.1007/3-7643-7449-7 2006
[3]

Frankland, 2-track algebras and the Adams spectral sequence,Journal of Homotopy and Related Structures11(2016), 679–713, DOI:https://doi.org/10.1007/s40062-016-0147-x

H.-Joachim Baues and M. Frankland, 2-track algebras and the Adams spectral sequence,Journal of Homotopy and Related Structures11(2016), 679–713, DOI:https://doi.org/10.1007/s40062-016-0147-x

work page doi:10.1007/s40062-016-0147-x 2016
[4]

Jibladze, Secondary derived functors and the Adams spectral sequence,Topology45 (2006), 295–324, DOI:https://doi.org/10.1016/j.top.2005.08.001

H.-Joachim Baues and M. Jibladze, Secondary derived functors and the Adams spectral sequence,Topology45 (2006), 295–324, DOI:https://doi.org/10.1016/j.top.2005.08.001

work page doi:10.1016/j.top.2005.08.001 2006
[5]

R. R. Bruner and J. Rognes, The Adams spectral sequence for the image-of-Jspectrum,Transactions of the American Mathematical Society375(2022), 5803–5827, DOI:https://doi.org/10.1090/tran/8680. ArXiv:2105.02601v3, arXiv DOI:https://doi.org/10.48550/arXiv.2105.02601

work page doi:10.1090/tran/8680 2022
[6]

R. R. Bruner, The fiber of Sq n, arXiv:2601.04028v2 (2026), arXiv DOI:https://doi.org/10.48550/arXiv. 2601.04028. To appear in Homology, Homotopy and Applications

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv 2026
[7]

D. E. Chua, TheE 3 page of the Adams spectral sequence, arXiv:2105.07628v2 (2022); doctoral dissertation, Harvard University Graduate School of Arts and Sciences, 2022, arXiv DOI:https://doi.org/10.48550/ arXiv.2105.07628

work page arXiv 2022
[8]

D. M. Davis, The cohomology of the spectrumbJ,Bolet´ ın de la Sociedad Matem´ atica Mexicana(2)20(1975), 6–11, available athttps://www.boletin.math.org.mx/pdf/2/20/BSMM(2).20.6-11.pdf

work page 1975
[9]

Milnor, The Steenrod algebra and its dual,Annals of Mathematics(2)67(1958), 150–171, DOI:https: //doi.org/10.2307/1969932

J. Milnor, The Steenrod algebra and its dual,Annals of Mathematics(2)67(1958), 150–171, DOI:https: //doi.org/10.2307/1969932. SECONDARY EXT OF THE FIBER OFSq n 19

work page doi:10.2307/1969932 1958
[10]

Nassau, On the secondary Steenrod algebra,New York Journal of Mathematics18(2012), 679–705, available athttps://nyjm.albany.edu/j/2012/18-37v.pdf

C. Nassau, On the secondary Steenrod algebra,New York Journal of Mathematics18(2012), 679–705, available athttps://nyjm.albany.edu/j/2012/18-37v.pdf. Department of Mathematics, FPT University, Quy Nhon AI Campus, An Phu Thinh New Urban Area, Vietnam Email address:dangphuc150488@gmail.com

work page 2012

[1] [1]

J. F. Adams, On the structure and applications of the Steenrod algebra,Commentarii Mathematici Helvetici 32(1958), 180–214, DOI:https://doi.org/10.1007/BF02564578

work page doi:10.1007/bf02564578 1958

[2] [2]

247, Birkh¨ auser Basel, 2006, DOI:https://doi.org/10.1007/3-7643-7449-7

H.-Joachim Baues,The Algebra of Secondary Cohomology Operations, Progress in Mathematics, vol. 247, Birkh¨ auser Basel, 2006, DOI:https://doi.org/10.1007/3-7643-7449-7

work page doi:10.1007/3-7643-7449-7 2006

[3] [3]

Frankland, 2-track algebras and the Adams spectral sequence,Journal of Homotopy and Related Structures11(2016), 679–713, DOI:https://doi.org/10.1007/s40062-016-0147-x

H.-Joachim Baues and M. Frankland, 2-track algebras and the Adams spectral sequence,Journal of Homotopy and Related Structures11(2016), 679–713, DOI:https://doi.org/10.1007/s40062-016-0147-x

work page doi:10.1007/s40062-016-0147-x 2016

[4] [4]

Jibladze, Secondary derived functors and the Adams spectral sequence,Topology45 (2006), 295–324, DOI:https://doi.org/10.1016/j.top.2005.08.001

H.-Joachim Baues and M. Jibladze, Secondary derived functors and the Adams spectral sequence,Topology45 (2006), 295–324, DOI:https://doi.org/10.1016/j.top.2005.08.001

work page doi:10.1016/j.top.2005.08.001 2006

[5] [5]

R. R. Bruner and J. Rognes, The Adams spectral sequence for the image-of-Jspectrum,Transactions of the American Mathematical Society375(2022), 5803–5827, DOI:https://doi.org/10.1090/tran/8680. ArXiv:2105.02601v3, arXiv DOI:https://doi.org/10.48550/arXiv.2105.02601

work page doi:10.1090/tran/8680 2022

[6] [6]

R. R. Bruner, The fiber of Sq n, arXiv:2601.04028v2 (2026), arXiv DOI:https://doi.org/10.48550/arXiv. 2601.04028. To appear in Homology, Homotopy and Applications

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv 2026

[7] [7]

D. E. Chua, TheE 3 page of the Adams spectral sequence, arXiv:2105.07628v2 (2022); doctoral dissertation, Harvard University Graduate School of Arts and Sciences, 2022, arXiv DOI:https://doi.org/10.48550/ arXiv.2105.07628

work page arXiv 2022

[8] [8]

D. M. Davis, The cohomology of the spectrumbJ,Bolet´ ın de la Sociedad Matem´ atica Mexicana(2)20(1975), 6–11, available athttps://www.boletin.math.org.mx/pdf/2/20/BSMM(2).20.6-11.pdf

work page 1975

[9] [9]

Milnor, The Steenrod algebra and its dual,Annals of Mathematics(2)67(1958), 150–171, DOI:https: //doi.org/10.2307/1969932

J. Milnor, The Steenrod algebra and its dual,Annals of Mathematics(2)67(1958), 150–171, DOI:https: //doi.org/10.2307/1969932. SECONDARY EXT OF THE FIBER OFSq n 19

work page doi:10.2307/1969932 1958

[10] [10]

Nassau, On the secondary Steenrod algebra,New York Journal of Mathematics18(2012), 679–705, available athttps://nyjm.albany.edu/j/2012/18-37v.pdf

C. Nassau, On the secondary Steenrod algebra,New York Journal of Mathematics18(2012), 679–705, available athttps://nyjm.albany.edu/j/2012/18-37v.pdf. Department of Mathematics, FPT University, Quy Nhon AI Campus, An Phu Thinh New Urban Area, Vietnam Email address:dangphuc150488@gmail.com

work page 2012