On periodic families in the stable stems of height two

Christian Carrick; Jack Morgan Davies

REVIEW 3 major objections 5 minor 2 cited by

On periodic families in the stable stems of height two

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper proves that the 2-primary stable homotopy groups of spheres contain 125 nonvanishing $v_2^{32}$-periodic families, all with zero image in topological modular forms, and derives exotic spheres in three new congruence classes of…

desk verdict A serious, explicitly presented computation of 125 v2-periodic families that deserves referee time, but whose nonvanishing proof leans on differentials imported from an unverified preprint by overlapping authors. read the letter →

arxiv 2506.20507 v2 pith:GORBAEDR submitted 2025-06-25 math.AT math.KT

Christian Carrick , Jack Morgan Davies This is my paper · ORCID

classification math.ATmath.KT MSC 55Q4555T1555Q5155P42

keywords stablehomotopygroupsofspheresperiodicfamiliestopologicalmodularformssyntheticspectraAdams–NovikovspectralsequenceAtkin–Lehnerinvolutionexoticv2-periodic

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

The paper proves Theorem A: the 2-primary stable homotopy groups of spheres contain 125 nonvanishing $v_2^{32}$-periodic families, distributed over nineteen congruence classes of degrees modulo 192, with orders as listed in Table 1 and with all generators mapping to zero in the homotopy of topological modular forms (TMF). The families are detected instead in the equalizer $J_0(3)$ of the Atkin–Lehner inflation map and the canonical map from TMF to $\mathrm{TMF}_0(3)$, so they are invisible to TMF yet survive in the sphere. The computation reconfirms and refines previously known families from earlier papers and adds 50 families not previously in the literature. As a corollary, exotic spheres exist in all dimensions congruent to 72, 144, and 168 modulo 192, with very exotic spheres in dimensions 143, 145, and 169 modulo 192.

What carries the argument

The load-bearing mechanism is the deleting-differentials technique in the category of BP-synthetic spectra. Given a fibre sequence of synthetic spectra $F \to X \to Y$, if a class $b \in \pi_{*,*} X$ has $\bar{b} \neq 0$ and every possible source $a$ of a differential $d_r(\bar{a}) = \bar{b}$ maps to $f(a) \neq 0$, then any lift of $b$ to $F$ is $\tau^{r-1}$-torsion free; applied to $X = \mathrm{TMF}_{BP}$, $Y = \mathrm{TMF}_0(3)_{BP}$, and $f = q - p$ with the Atkin–Lehner twist $q = w \circ p$, this deletes the differentials that would otherwise kill the target classes in $\pi_* \mathrm{TMF}$. The equalizer $J_0(3)_{BP}$ of $p$ and $q$ is the detection spectrum, and the paper shows most of Table 1's classes are nonzero in its image.

What would settle it

Independently compute the 2-primary descent spectral sequence for TMF in the range of Table 3 and check the listed differentials, for example $d_5((4k+1)\Delta) = \nu\bar{\kappa}$, $d_9(\eta\Delta^2) = \varepsilon\bar{\kappa}^2$, and $d_9(\eta\Delta^3) = \bar{\kappa}^2[\varepsilon\Delta]$. If any of these fails, or if the source of a listed differential maps to zero under $q-p$ in the Adams–Novikov spectral sequence for $\mathrm{TMF}_0(3)$, then the corresponding row of Table 1 need not contribute a nonzero family, and the count of 125 would be too high.

Watch

Extended reading notes

Core claim

The central claim is that Table 1 lists 125 nonvanishing $v_2^{32}$-periodic families in $\pi_d S_2$ for each degree modulo 192 shown, with the stated orders and with trivial image in $\pi_* \mathrm{TMF}$. The proof produces the families by lifting each generator to the synthetic Hurewicz image of $\mathrm{TMF}_{BP}$, where the known descent-spectral-sequence differentials for TMF kill it, and then using a deleting-differentials argument to show those differentials cannot lift to the sphere; the target instead survives in the equalizer $J_0(3)_{BP}$ of the two maps $\mathrm{TMF}_{BP} \to \mathrm{TMF}_0(3)_{BP}$. A smaller set of families, detected only in the sphere, is handled by a filtration argument using the 1-line of the Adams–Novikov spectral sequence. Corollary B translates the surviving families into exotic spheres in the stated congruence classes.

Load-bearing premise

The load-bearing premise is that the differentials in the 2-primary descent spectral sequence for TMF listed in [CDvN24a, Section 6] (Theorem 5.2, Table 3) are computed correctly; the paper does not reproduce those computations, and a single misidentified differential would let the corresponding families be killed in the sphere rather than survive.

Editorial extensions

If this is right

Table 1 yields 125 nonvanishing $v_2^{32}$-periodic families in $\pi_* S_2$, of which 50 are new and the rest reconfirm known families from [BHHM20], [BBQ24], and [BQ24].
All 125 families vanish in $\pi_* \mathrm{TMF}$ but are detected in the equalizer $J_0(3)$, so TMF's classical Hurewicz image is not the only source of $v_2$-periodic phenomena at the prime 2.
Corollary B: exotic spheres exist in all dimensions congruent to 72, 144, and 168 modulo 192, and very exotic spheres in dimensions 143, 145, and 169 modulo 192.
The same techniques reconfirm the families tentatively suggested in the earlier brief report [DFHH14, §15], including the classes in degrees 47 and 48, and confirm the nonvanishing of the family assembled from $\bar{\kappa}^6$ in degree 120.
Because TMF and $J_0(3)$ are MU-nilpotent, the proof also shows all listed families have nonzero image in the $K(2)$-local sphere, while the method cannot produce nonzero families in the $T(2)$-local sphere that vanish $K(2)$-locally.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the same equalizer construction is applied to the level-five and level-seven analogues $\mathrm{TMF}_0(5)$ at $p=2$ and $\mathrm{TMF}_0(7)$ at $p=3$, using the level-specific computations referenced in the paper, the deleting-differentials argument could plausibly produce further simple torsion families beyond these 125; the paper leaves this open in Question 5.10.
A systematic computation of the synthetic Hurewicz image of $\mathrm{TMF}_{BP}$, which the paper does only case-by-case, would likely extend the method to unresolved classes such as $\bar{\kappa}[\nu\Delta^4]$, $\bar{\kappa}[2\nu\Delta^5]$, and $\bar{\kappa}[\nu\Delta^6]$ in degrees 119, 143, and 167, as suggested in Question 5.6.
The fact that several rows of Table 1 are detected only by a filtration argument rather than by $J_0(3)$ suggests that other fixed-point spectra built from TMF by Hecke-type operations could detect additional families that $J_0(3)$ misses, including the 2-torsion family in degree 122 and the 8-torsion family in degree 170 mentioned in Question 5.11.

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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

The paper you want to know about: Carrick and Davies claim 125 nonzero v2^32-periodic families in the 2-primary stable stems, all vanishing in TMF, with many detected by a height-2 analogue of the image-of-J spectrum. If Theorem A holds, this is the biggest progress on v2-periodic families at the prime 2 in years, and it settles a batch of Hopkins–Mahowald predictions. That's the headline.

What is genuinely new: the 50 families not previously in the literature, and the method. The deleting-differentials technique in synthetic spectra, combined with the Atkin–Lehner involution on TMF_0(3), is a clever and portable way to rule out lifts of TMF differentials to the sphere. The paper is unusually explicit: Table 1 and Table 2 give generators, orders, and how the counting works, and §5.3 walks through every row. The recovery of the known families of BHHM20, BBQ24, BQ24 is a sensible sanity check.

Where I have a soft spot, it's the one the stress-test flags. The nonvanishing of the checked families in Theorem 5.2 depends on the differentials in Table 3, and those are not proved here—they are quoted from [CDvN24a, §6], a preprint by overlapping authors. One wrong row, e.g., the d5 in degree 47, and the affected families disappear. That is a real verification gap, not an internal inconsistency. It doesn't make me think the result is false; the differentials in the TMF DSS are heavily constrained and the authors clearly know them. But for a paper whose main theorem is a list of nonzero classes, I want the referee to spend time on those differentials before accepting.

There are also smaller places where the proof leans on inspection of the [IWX22] charts, and a few assertions like 'the blue dot indicates...' that are not fully spelled out. That is normal for this literature, but it adds to the self-containedness problem.

Bottom line: this is a serious, well-written paper by people who know what they are doing. It deserves a serious referee—two of them, ideally, one of whom checks the TMF DSS differentials against [CDvN24a]. I would not desk-reject it. I would not cite it as established fact until [CDvN24a] is also verified, but I would cite it as a preprint and bring it to reading group.

Referee Report

3 major / 5 minor

Summary. The paper develops a method based on BP-synthetic spectra and the Atkin–Lehner involution on TMF_0(3) to detect v_2^32-periodic families in the 2-primary stable homotopy groups of spheres that vanish in TMF. The main theorem (Theorem A) lists 125 nonzero v_2^32-periodic families in π_* S_2, all with trivial image in π_* TMF, and recovers/confirms previously known families of Behrens–Hill–Hopkins–Mahowald, Bhattacharya–Bobkova–Quigley, and Bobkova–Quigley. Corollary B derives existence of exotic spheres in dimensions congruent to 72, 144, and 168 modulo 192, and very exotic spheres in dimensions congruent to 143, 145, and 169 modulo 192. The proof has two main ingredients: an analysis of the synthetic Hurewicz image of TMF (Section 4) and a 'deleting differentials' argument using the detection spectrum J_0(3) (Sections 3 and 5).

Significance. If the main theorem is correct, this is a substantial contribution to the computation of v_2-periodic families in the 2-primary stable stems. It provides the first unified and systematic detection of many v_2-periodic families that are invisible to TMF, confirms a number of Hopkins–Mahowald predictions, and yields new exotic sphere existence results. The paper is unusually explicit: it gives detailed tables of generators (Tables 1 and 2), differentials (Table 3), and a careful treatment of the ambiguity in choosing periodic families (Theorem 2.2). The synthetic-spectra framework is used elegantly, and the central detection criterion (Corollary 3.3) is clearly formulated. The main risk is the heavy reliance on external, not-yet-published computations for the TMF descent spectral sequence.

major comments (3)

[§5.2, Table 3 and Theorem 5.2] The deleting-differentials argument (Corollary 3.3) is the engine of the proof of Theorem A, and it requires, for each target x in Table 1, that x is hit by exactly the stated differential d_r in σ(TMF^BP) and that the source is distinguished by q-p. These differentials are not proved in this paper; they are quoted from the preprint [CDvN24a, §6]. The proof of Theorem 5.2 states: 'references to all of these differentials can be found on their appropriate pages in [CDvN24a, §6]', and Table 3 records only the final differentials. Since [CDvN24a] is a preprint by overlapping authors and is not reproduced or independently checked here, the correctness of every row of Table 3 is an unverified premise. A single wrong row does not just lose one class: for example, the degree-47 row d5((2k+1)Δ^2)=2ν̄κΔ underlies families 47a and 47b; if this d5 is absent or has a different source, then [2ν̄κΔ] need not be τ-torsion free in S^BP and need not vanish in π_*TMF. The same structure recurs for each ✓ row. Hence Theorem A's '125 families ... with trivial image in TMF' is conditional on external, unverified differential data. I recommend that the authors prove these differentials (or provide a detailed verification) within the paper, or clearly state the theorem as conditional on the acceptance of the companion preprint.
[§4.2–§4.4] The proof of Theorem 4.2, which establishes the synthetic Hurewicz image of the candidate classes, depends on a number of claims justified by 'inspection of the charts of [IWX22]'. This includes Lemma 4.12 (stem 47), Lemma 4.18–4.20 (stem 71), and Lemma 4.21–4.23 (higher stems). These inspections involve subtle points, such as the τ-power torsion nature of the class l1 and the hidden η-extension from (71,7) to (72,10) in the proof of Lemma 4.18. Since the charts are publicly available, this is not an error, but the paper should provide precise bidegree data (e.g., the exact classes and their coordinates) for each inspection claim so that the reader can verify them without re-deriving the entire charts. This is load-bearing because these classes are the input to the detection argument; a misidentified chart class would invalidate the corresponding family.
[§5.2, proof of Theorem 5.2] The application of Corollary 3.3 requires, for each row of Table 3, not only that the stated d_r is deleted, but also that the groups π_{s+1,f-r-i}J_0(3)^BP/τ vanish for i≥1 so that the lifted classes are permanent cycles. The proof asserts this follows from Lemmas 3.12–3.14, but no systematic verification is given for the 14 degree rows. Since the checkerboard pattern and negative-filtration vanishing make this a finite check, I request a table or a clear argument covering all bidegrees involved.

minor comments (5)

[Throughout] The paper refers to 'Equation (5.1)' and 'Equation (5.2)' when it means Theorem 5.1 and Theorem 5.2, and similarly 'Equation (4.10)' for Proposition 4.10. There are no displayed equation numbers in the text, so these references should be corrected for clarity.
[§1.3 and §3.2.1] There are typos: 'paralellisable' should be 'parallelisable', and 'prefered' should be 'preferred'.
[Definition 2.1] In the phrase 'admits v_k^h-self map v on F', the symbol F is undefined; it should likely be 'on M'.
[Table 3] The row for degree 143 reads 'd5(Δ^6)=η̄κη∆^6', which appears to have a duplicated symbol; please check the notation.
[References [IWX20a], [IWX22]] The references to the Adams–Novikov and Adams charts should specify the version and date used, since these are living documents.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction; Theorem A is a genuine detection result whose load-bearing inputs are external or separately computed.

full rationale

The derivation chain is not circular. The nonvanishing families are proved in two independent stages. Stage 1 (Section 4) identifies candidate classes in the synthetic Hurewicz image of TMF^BP; the bulk of these come from the classical Hurewicz image of tmf computed by Behrens–Mahowald–Quigley [BMQ23], an external published benchmark, with the exceptional degree-47, -71, and -95 classes handled by explicit F2-synthetic Moore-spectrum and Toda-bracket arguments (Propositions 4.10, 4.17, 4.23) against the published [IWX20a, IWX22] charts. Stage 2 (Section 5) uses Corollary 3.3, a general deleting-differentials principle, to show that these classes are τ-torsion free in S^BP. The only part of Stage 2 that cites overlapping prior work is the list of d_r differentials in Table 3, whose targets are the TMF-ANSS representatives of the classes; the proof states 'references to all of these differentials can be found on their appropriate pages in [CDvN24a, §6]'. This is a genuine verification dependency on a preprint by the same authors, and a wrong row in Table 3 could invalidate the corresponding families, but it is not a circularity in the sense of this review: [CDvN24a] computes the descent spectral sequence for TMF, and that computation does not assume or restate the sphere-nonvanishing claim of Theorem A. The Atkin–Lehner involution used to build J0(3) is from the published [Dav24a]. No fitted parameters are introduced, and no quantity being 'predicted' is used as an input by construction. Thus the central claim has independent content; the appropriate finding is no significant circularity, with the [CDvN24a] dependency recorded as a verification risk rather than a circular step.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim does not depend on any fitted free parameters. It does depend on a stack of prior computational inputs: the periodicity theorem for v_h self maps, the computed DSS/ANSS for TMF from the authors' own preprint [CDvN24a], the classical Hurewicz image of tmf [BMQ23], the Atkin-Lehner involution [Dav24a], and the Isaksen-Wang-Xu charts [IWX20a, IWX22]. These are domain assumptions, not ad hoc inventions. No new entities are introduced beyond a limit construction J0(3) built from known spectra.

assumptions (5)

standard math The periodicity theorem and the uniqueness of v_h self maps (Hopkins-Smith, Ravenel) hold.
Used in Definition 2.1 and Theorem 2.2 to guarantee that different periodic families with the same generator agree cofinally.
domain assumption The 2-primary descent spectral sequence for TMF has been computed correctly, as in [CDvN24a].
The differentials listed in Table 3 (Theorem 5.2), which are needed to show candidate classes vanish in TMF and to delete those differentials, are delegated to the preprint [CDvN24a].
domain assumption The classical Hurewicz image of tmf is as computed by Behrens-Mahowald-Quigley in [BMQ23].
Used in Theorem 4.2 to identify which classes in pi_* TMF lift from the sphere; many periodicity generators are asserted to be v_2^32-periodic by [BMQ23].
domain assumption The Atkin-Lehner involution w on TMF0(3) exists as an E8-ring map with the stated action on the DSS, as constructed by Davies in [Dav24a].
The detection spectrum J0(3) of Definition 3.8 depends on this map; its properties, for instance w^2 equivalent to psi_N in Lemma 3.14, are cited from [Dav24a].
domain assumption The Adams-Novikov spectral sequence charts of Isaksen-Wang-Xu [IWX20a, IWX22] are accurate.
Multiple steps in Sections 4.3 and 4.4, for example Lemmas 4.12, 4.18 and 4.20, rely on inspection of these charts to identify extensions and torsions.

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Cite this review

Pith. "Pith review of On periodic families in the stable stems of height two." pith.science (2026). https://pith.science/paper/GORBAEDR

@misc{pith2026250620507,
  author       = {Pith},
  title        = {Pith review of: On periodic families in the stable stems of height two},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GORBAEDR}},
  note         = {Machine review of arXiv:2506.20507}
}

read the original abstract

We discover a host of infinite periodic families in the 2-primary stable homotopy groups of spheres. We also confirm the existence of many families predicted by Hopkins--Mahowald. These families appear in nineteen different congruence classes of degrees modulo 192, seven of them consist of simple 4-torsion elements, and another four of simple 8-torsion. They all vanish in the homotopy groups of the spectrum TMF of topological modular forms, but we show that they are detected in the fixed-points of TMF with respect to an Atkin--Lehner involution. As a consequence, we confirm the existence of exotic spheres in all dimensions congruent to 72, 144, and 168 modulo 192.

Figures

Figures reproduced from arXiv: 2506.20507 by the authors.

**Figure 1.** ANSS for KO on the left and the signature of fibpνψ3´ 1: ν KO Ñ ν KOq on the right. The blue classes are lifts from the ANSS for KO, the red classes lie in the image of the boundary map, and the group in bidegree p3, 1q is Z{8Z; also see [CD24b, Fig.12]. This argument then shows that η 3 ‰ 0 P π˚ fibpψ 3 ´ 1q, and thus η 3 ‰ 0 P π˚S. We state now an almost tautologically general case of deleted differentials. In the… view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Revisiting the $\beta_1$-action on the $3$-primary stable homotopy groups of spheres
math.AT 2025-11 conditional novelty 6.0 of 10

Products of the 3-primary β₁-periodic family in the stable homotopy groups of spheres are nonzero for up to five factors and zero for six or more, with analogous sharp cutoffs for β₂-, α₁β₂-, and [α₁β₃/3]-twisted products.
Periodic phenomena in stable motivic homotopy theory
math.AT 2026-07 unverdicted novelty 2.0 of 10

A survey of periodic phenomena in stable motivic homotopy theory, organizing known motivic Adams spectral sequence computations and open problems; no new theorem is proven.

Reference graph

Works this paper leans on

16 extracted references · 13 canonical work pages · cited by 2 Pith papers

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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Nonvanishing of products in $v_2$-periodic families at the prime $3$

[CD24a] Christian Carrick and Jack Morgan Davies. Nonvanishing of products inv 2-periodic families at the prime3. Preprint, arXiv:2410.02564 [math.AT] (2024),

work page Pith review arXiv 2024

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The descent spectral sequence for topological modular forms

accepted in Transactions of the American Mathematical Society. [CDvN24a] ChristianCarrick, JackMorganDavies, andSvenvanNigtevecht.Thedescentspectral sequence for topological modular forms. Preprint, arXiv:2412.01640 (2024),

work page Pith review arXiv 2024

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Descent spectral sequences through synthetic spectra

[CDvN24b] Christian Carrick, Jack Morgan Davies, and Sven van Nigtevecht. Descent spectral sequences through synthetic spectra. Preprint, arXiv:2407.01507 [math.AT] (2024),

work page Pith review arXiv 2024

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[Hop95] Michael J. Hopkins. Topological modular forms, the Witten genus, and the theorem of the cube. InProceedings of the international congress of mathematicians, ICM ’94, 30 CHRISTIAN CARRICK AND JACK MORGAN DA VIES August 3-11, 1994, Zürich, Switzerland. Vol. I, pages 554–565. Basel: Birkhäuser,

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Isaksen, Guozhen Wang, and Zhouli Xu

[IWX22] Daniel C. Isaksen, Guozhen Wang, and Zhouli Xu. Classical andC-motivic Adams charts, v1. Available athttps://doi.org/10.5281/zenodo.6987157, October

work page doi:10.5281/zenodo.6987157

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The homotopy groups of the spectrum Tmf

[Kon12] Johan Konter. The homotopy groups of the spectrum Tmf. available at arXiv: 1212.3656,

work page Pith review arXiv

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Computation of the homotopy of the spectrum tmf

[Bau08] Tilman Bauer. Computation of the homotopy of the spectrum tmf. InProceedings of the conference on groups, homotopy and configuration spaces, University of Tokyo, Japan, July 5–11, 2005 in honor of the 60th birthday of Fred Cohen, pages 11–40. Coventry: Geometry & Topology Publications,

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