Pith. sign in

REVIEW 2 major objections 4 minor 43 references

On higher real $K$-theories and finite spectra

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The connective higher real K-theory spectra $eo_h$ are fp spectra of chromatic type $h$ at every height, and this forces any existing generalized Moore spectrum $\mathbb{S}/(2^{i_0},v_1^{i_1},\ldots,v_h^{i_h})$ to satisfy $\nu_2(\prod…

desk verdict The fp-ness theorem for higher real K-theories is real and new; the paper is referee-ready after fixing a dimension-formula indexing error and pinning down the Burklund–Levy dependency in Theorem 1.3. read the letter →

arxiv 2507.07051 v1 pith:Z3PQLLEI submitted 2025-07-09 math.KT math.AT

Christian Carrick , Michael A. Hill This is my paper · ORCID
classification math.KTmath.AT MSC 55N2255P4255P91
keywords higherrealK-theoryfpspectrachromaticheightgeneralizedMoorealgebraicofequivariantstablehomotopytheorytruncatedBrown-PetersonKoszulfiltration
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

This paper tries to establish that the connective higher real $K$-theory spectra $eo_h$---fixed points of equivariant truncated Brown--Peterson spectra $BP^{(G)}\langle m\rangle$ with $h=m|G|/2$---are fp spectra of chromatic type $h$ for every height $h$. That means they are bounded below, $2$-complete, have finitely presented cohomology over the Steenrod algebra, and their smash products with finite spectra stay finite exactly above chromatic height $h$. Such spectra are workable, $ko$-like replacements for $BP\langle h\rangle$ at heights where almost nothing explicit was known. The proof is carried by an equivariant Koszul filtration, and the same filtration feeds an Euler characteristic computation making $\chi_{BP\langle h\rangle}$ divisible by $2^n$ when $h=2^{n-1}m$. The advertised payoff is a new prime-$2$ obstruction: if a generalized Moore spectrum $\mathbb{S}/(2^{i_0},v_1^{i_1},\ldots,v_h^{i_h})$ exists, then $\nu_2(\prod i_k)>\nu_2(h)$. This is a partial verification of a conjecture about the algebraic $K$-theory of finite type-$h$ spectra.

What carries the argument

The load-bearing mechanism is a normed Koszul filtration for equivariant quotients. The paper replaces the non-existent cofiber sequence $BP^{(G)}\langle m-1\rangle\to BP^{(G)}\langle m\rangle$ with a filtration whose associated graded pieces are indexed by $C_2$-equivariant functions $f\colon G\to\{0,1\}$, presenting each layer as an induced norm of a quotient by a proper subgroup's generators; this mirrors the Koszul resolution of a residue field over a polynomial ring. The algebraic input that closes the induction is that $\pi_*^e BP^{(G)}\langle m\rangle/(2,v_1,\ldots,v_h)$ is a finite $\mathbb{F}_2$-vector space of odd dimension, computed as $\prod_{j=0}^{|G|/2-1}\binom{jm}{m}_2$. On the K-theory side, the same filtration yields the torsion relation between fixed-point classes and identifies $\chi_{BP^{(G)}\langle m\rangle^e}$ with that odd Gaussian-binomial factor times $\chi_{BP\langle h\rangle}$; a cited rational-generation theorem then transfers the divisibility to the Moore-spectrum obstruction. The Euler characteristic itself is $\chi_E(K)=\sum_i(-1)^i\log_2|\pi_i(E\otimes K)|$, pairing an fp spectrum $E$ with a finite spectrum $K$ of higher type.

What would settle it

Smash $eo_3 = BP^{(C_2)}\langle 3\rangle^{C_2}$ with a known finite type-$3$ complex such as $Z/v_2$ and compute its homotopy groups; the paper predicts infinitely many nonzero $2$-torsion groups in the answer, so finding only finitely many nonzero groups would refute the central fp-type claim.

Watch

Extended reading notes

Core claim

The central claim, Theorem 2.5, is that for every subgroup $H$ of a cyclic $2$-group $G=C_{2^n}$, the fixed-point spectrum $BP^{(G)}\langle m\rangle^H$ is an fp spectrum of type $m|G|/2$; in particular $eo_h(H)$ has fp type $h$. Concretely, these spectra are bounded below, $2$-complete, and finitely presented as modules over the Steenrod algebra, and they detect exactly the chromatic height-$h$ layer: smashing with a finite spectrum of type $h+1$ gives finite homotopy, while smashing with a finite type-$h$ spectrum does not. From this the paper derives the K-theory relation $|H|[BP^{(G)}\langle m\rangle^H]\equiv [BP^{(G)}\langle m\rangle^e]$ modulo torsion in $K_0(fp_{\le h})$, which makes the Euler characteristic $\chi_{BP\langle h\rangle}$ divisible by $2^n$ whenever $h=2^{n-1}m$. Theorem 1.2 follows: existence of $\mathbb{S}/(2^{i_0},v_1^{i_1},\ldots,v_h^{i_h})$ forces $\nu_2(\prod_{k=0}^h i_k)>\nu_2(h)$. The proofs proceed by induction on height, using the transchromatic height-shifting layers in the equivariant slice filtration.

Load-bearing premise

The argument from fp spectra to nonexistence rests on a cited, still-unpublished theorem saying that a certain integer-valued counting invariant rationally detects, up to odd multiples, all finite spectra above height $h$; if that theorem is false or secretly assumes the Moore spectrum already exists, the divisibility and nonexistence results collapse.

Editorial extensions

If this is right

  • For every height $h=2^{n-1}m$ with $h\not\equiv 2\bmod 4$, the spectrum $eo_h$ is bounded below, $2$-complete, and finitely presented over the Steenrod algebra, so Adams-spectral-sequence methods previously limited to $ko$ and $tmf$ become available at arbitrary height.
  • The Euler characteristic $\chi_{BP\langle h\rangle}$ is divisible by $2^n$, confirming the divisibility half of the conjecture that its image is generated by the order of a maximal finite $2$-subgroup of the height-$h$ stabilizer group.
  • No generalized Moore spectrum $\mathbb{S}/(2^{i_0},v_1^{i_1},\ldots,v_h^{i_h})$ can exist unless $\nu_2(i_0\cdots i_h)>\nu_2(h)$, giving the first prime-$2$ nonexistence criterion valid at all heights.
  • For these spectra, finite-height chromatic localization $L_f^n$ agrees with $L_n$, so the telescope-conjecture condition holds in this class, and the completion map $BP^{(G)}\langle m\rangle^H\to BP^{(G)}\langle m\rangle^{hH}$ has bounded-above fiber.
  • The connective cover of the Borel completion $\tau_{\ge0}(BP^{(G)}\langle m\rangle^{hH})$ is again an fp spectrum of type $h$.

Reading between the lines

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

  • Beyond the paper's cyclic-group setting, the filtration method is built from the norm functor for cyclic $2$-groups; adapting it to a non-cyclic maximal subgroup such as $Q_8$ would fill the gap at heights $h\equiv 2\bmod 4$, where the paper leaves the divisibility question open.
  • The odd Gaussian-binomial factor separating $\chi_{BP^{(G)}\langle m\rangle^e}$ from $\chi_{BP\langle h\rangle}$ suggests a geometric reading: if these coefficients count degrees of forgetful covers on a moduli stack of formal groups with group actions, the slice layers would acquire a modular interpretation predicting the structure of higher fixed-point spectra.
  • A natural next step is to compute the minimal $v_h$-periodicity of $eo_h$ from the equivariant slice spectral sequence; if the minimal exponent exceeds the valuation bound, Theorem 1.2 could be upgraded from a divisibility statement to an exponent-by-exponent nonexistence criterion.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper introduces the higher truncated Brown-Peterson spectra BP^{(G)}⟨m⟩ and their fixed points eo_h(H), and proves that these are fp spectra in the sense of Mahowald-Rezk, of chromatic type h = m|G|/2. The main structural result, Theorem 2.5, is obtained by an equivariant induction using Koszul-type filtrations and a computation of the underlying homotopy modulo (2,v_1,...,v_h). The paper then uses Levy's Euler characteristic for fp spectra to derive K-theory relations among fixed points, identifies the Euler characteristic of BP^{(G)}⟨m⟩^e with an odd multiple of χ_{BP⟨h⟩}, and concludes that the image of χ_{BP⟨h⟩} is divisible by 2^n (Theorem 1.3). A corollary is the nonexistence criterion for generalized Moore spectra: if S/(2^{i_0},...,v_h^{i_h}) exists, then ν_2(∏ i_k) > ν_2(h) (Theorem 1.2). The final section proves Borel-completeness results for chromatic localizations of MU^{(G)}-modules.

Significance. If the central results hold, this is a substantial contribution to chromatic homotopy theory: it provides new explicit fp spectra at all heights, gives the first 2-primary nonexistence constraint on generalized Moore spectra valid at all heights, and partially verifies Levy's conjecture on the image of χ_{BP⟨h⟩}. The paper is careful and self-contained in its induction in Sections 4-5, and the identification of the Poincaré series and Gaussian binomial coefficients is explicit and falsifiable. The main external risk is the dependence of Section 6 on the unpublished Burklund-Levy theorem [12]; this is acknowledged but must be resolved before the paper can be judged. A second, local but load-bearing issue is the off-by-one indexing in Corollary 3.8, which is easily corrected.

major comments (2)
  1. [Corollary 3.8] The displayed formula for the dimension of π_e^*BP^{(G)}⟨m⟩/(2,v_1,...,v_h) has an off-by-one indexing error: the product is written over j=0,...,|G|/2−1 of the Gaussian binomial coefficient [jm;m]_2, and by Definition 3.6 the j=0 term [0;m]_2 is zero for m>0. This makes the stated dimension zero and contradicts Theorem 2.6 as well as the asserted oddness. The limiting computation in the proof appears to intend the product over j=1,...,|G|/2 of [jm;m]_2 (equivalently, shifting the index in the displayed product). Because this odd multiplier is used in Proposition 6.11 to identify χ_{BP^{(G)}⟨m⟩^e} with an odd multiple of χ_{BP⟨h⟩}, the indexing error is load-bearing and must be corrected.
  2. [Theorem 6.9 and Proposition 6.11] The proof of the global divisibility statement Theorem 1.3 and of the second part of Proposition 6.11 depends essentially on the unpublished Burklund-Levy theorem [12]. Theorem 6.9 is stated as a direct consequence of [12], but the needed assertion is not merely that χ_{BP⟨h⟩} is a rational isomorphism; it is that any generalized Moore spectrum rationally generates K_0(Sp^ω_{>h})[1/2]. As written, the short proof of Theorem 6.9 is valid only if [12] contains exactly this rational-generation statement, and the manuscript does not quote or prove it. If [12] is not yet available in final form, or if it proves a weaker statement, then Theorem 1.3 and the identification of χ_{BP^{(G)}⟨m⟩^e} with an odd multiple of χ_{BP⟨h⟩} are unsupported. The authors should either include a proof of the required rational-generation statement, cite a published version of [12], or explicitly mark Theorem 1.3 and the second part of Proposition 6.11 as conditional. Note also that Theorem 1.2 can be proved without [12] by combining Theorem 6.8 with the first part of Proposition 6.11, so the dependency can be localized.
minor comments (4)
  1. [Lemma 5.3] The first line begins with the typo "Kor all C2 ⊂ H ⊂ G"; this should read "For all".
  2. [Theorem 6.9] The statement contains typographical corruption of the generalized Moore spectrum: the expression "vih n" and "S/(2^{i_0},...,v^{i_h}_n)" should read S/(2^{i_0},...,v_h^{i_h}).
  3. [Corollary 3.8, proof] In the limiting computation, the displayed quotient (1-x^{2^{jm+i-1}})/(1-x^{2^{i-1}}) should have numerator exponent 2^{(j-1)m+i} in order to match the Gaussian binomial formula after taking the limit; clarifying this would remove ambiguity about the intended indexing.
  4. [Corollary 6.12] The proof would benefit from explicitly stating that F/v is a generalized Moore spectrum of chromatic type h+1, since this is the point at which Proposition 6.11 is applied.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation found: the fp/type theorem and K-theory relations are proved internally, and the only load-bearing external input, Burklund–Levy [12], is independently authored.

full rationale

The paper's central derivation is self-contained apart from the cited Burklund–Levy theorem [12]. Theorem 2.5 (BP^(G)<m>^H is fp of type m|G|/2) is proved by an internal induction: Theorem 3.1 establishes algebraic nilpotence in the underlying homotopy ring; Section 5 lifts this to spectrum-level nilpotence via Corollary 4.14 and Proposition 4.15, using only the published equivariant Balmer-spectrum results [4,5]; and Corollary 6.12 proves the type is exactly h using only Proposition 6.11(1) and Theorem 6.8. The K-theory relation of Theorem 6.8 is derived from the Koszul filtration of Section 4, not from any assumed conclusion. Proposition 6.11(1) is a direct computation of the Euler characteristic on a generalized Moore spectrum; its second statement, identifying χ_BP^(G)<m>^e with an odd multiple of χ_BP<h>, uses Theorem 6.9, which is quoted from Burklund–Levy [12] ('To appear'). That identification is the only path to the global divisibility statement Theorem 1.3, so Theorem 1.3 is conditional on an independent, unpublished theorem. This is a correctness/rigor risk, not circularity. The advertised nonexistence result Theorem 1.2 could be obtained directly from Theorem 6.8 and Proposition 6.11(1) without [12], although as written the paper routes it through Theorem 1.3. Self-citations to [6], [15], [23], [25], and [27] supply constructions and structural formulas that do not assume the present results and are externally published. No fitted parameter is renamed as a prediction, and no definition presupposes the target theorem.

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

No free parameters are fitted to data. The central claim rests on prior structural theorems in equivariant and chromatic homotopy theory plus the unpublished Burklund-Levy result. No new particles, forces, or invented spectrum-level entities are introduced; the spectra eo_h and BP^{(G)}<m> were defined in earlier work [23].

assumptions (5)
  • standard math Thick subcategory theorem and the Balmer spectrum classification of finite G-spectra identify thick tensor-ideals generated by finite H-spectra of the same chromatic type.
    Invoked in Lemma 5.3 to conclude inf_H^e(K) lies in the thick tensor-ideal generated by N_{C2}^H(inf_{C2}^e K).
  • standard math The recursive formulas of [6, Theorem 1.1] describe the images of the v_i generators and the behavior of the G-action on π_* BP^{(G)}.
    Used throughout Section 3 to prove nilpotence of the t_i' mod (2, v_1, ..., v_h), hence the finiteness and odd dimension results.
  • domain assumption Burklund-Levy [12]: χ_{BP<h>}: K0(Sp^ω_{>h}) → Z is a rational isomorphism, and any existing generalized Moore spectrum rationally generates the domain.
    Load-bearing for Theorem 6.9 and Proposition 6.11; the cited paper is unpublished, listed as 'To appear'.
  • standard math Periodicity theorem of Devinatz-Hopkins-Smith supplies v_h-self maps on finite type h spectra, and Burklund [11] supplies finite homotopy commutative ring spectra of arbitrary chromatic type.
    Used in Proposition 5.1 and Corollary 6.12 to construct the finite test spectra and the quotients F/v.
  • domain assumption Hahn-Wilson theorem: if X is fp of type h, then X → L^f_h X is an isomorphism on homotopy groups in sufficiently high degrees; and MU-modules satisfy the telescope conjecture at all heights.
    Used in Section 7 for the Borel completion and telescope conjecture statements.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On higher real $K$-theories and finite spectra." pith.science (2026). https://pith.science/paper/Z3PQLLEI

@misc{pith2026250707051,
  author       = {Pith},
  title        = {Pith review of: On higher real $K$-theories and finite spectra},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z3PQLLEI}},
  note         = {Machine review of arXiv:2507.07051}
}
abstract

We study higher chromatic height analogues $eo_h$ of the connective real $K$-theory spectrum $ko$. We show that $eo_h$ is an fp spectrum of type $h$ in the sense of Mahowald--Rezk. We use these to study an Euler characteristic for fp spectra introduced by Ishan Levy, and give a partial answer to a question of Levy regarding the algebraic $K$-theory of the category of finite type $h$ spectra. As a corollary, we prove that if the generalized Moore spectrum $\mathbb{S}/(2^{i_0},v_1^{i_1},\ldots,v_h^{i_h})$ exists, then the $2$-adic valuation of $\prod i_j$ must exceed that of the height $h$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

43 extracted references · 30 canonical work pages

  1. [12]

    Burklund and I

    R. Burklund and I. Levy. The K-theoretic telescope conjecture away fromp. To appear

    work page

  2. [1]

    A periodicity theorem in homological algebra

    J. F. Adams. “A periodicity theorem in homological algebra”. English.Proc. Camb. Philos. Soc. 62 (1966), pp. 365–377.issn: 0008-1981

    work page 1966

  3. [2]

    On the groups J(X). IV

    J. F. Adams. “On the groups J(X). IV”. English.Topology5 (1966), pp. 21–71.issn: 0040-

    work page 1966

  4. [3]

    K-theory and reality

    M. F. Atiyah. “K-theory and reality.” English.Q. J. Math., Oxf. II. Ser.17 (1966), pp. 367–

    work page 1966

  5. [4]

    The spectrum of the equivariant stable homotopy category of a finite group

    P. Balmer and B. Sanders. “The spectrum of the equivariant stable homotopy category of a finite group”.Invent. Math. 208.1 (2017), pp. 283–326. issn: 0020-9910,1432-1297. doi: 10.1007/s00222-016-0691-3

    doi:10.1007/s00222-016-0691-3 2017

  6. [5]

    The Balmer spectrum of the equivariant homotopy category of a finite abelian group

    T. Barthel, M. Hausmann, N. Naumann, T. Nikolaus, J. Noel, and N. Stapleton. “The Balmer spectrum of the equivariant homotopy category of a finite abelian group”.Invent. Math. 216.1 (2019), pp. 215–240. issn: 0020-9910,1432-1297. doi: 10.1007/s00222- 018- 0846-5

    doi:10.1007/s00222- 2019

  7. [6]

    Models of Lubin-Tate spectra via real bordism theory

    A. Beaudry, M. A. Hill, X. D. Shi, and M. Zeng. “Models of Lubin-Tate spectra via real bordism theory”. Adv. Math.392 (2021), Paper No. 108020, 58.issn: 0001-8708. doi: 10. 1016/j.aim.2021.108020

    work page arXiv 2021

  8. [7]

    On the existence of av32 2 -self map on M (1, 4) at the prime 2

    M. Behrens, M. Hill, M. J. Hopkins, and M. Mahowald. “On the existence of av32 2 -self map on M (1, 4) at the prime 2.” English. Homology Homotopy Appl. 10.3 (2008), pp. 45–84. issn: 1532-0073; 1532-0081/e

    work page 2008

Show all 43 references
  1. [8]

    The 2-primary Hurewicz image of tmf

    M. Behrens, M. Mahowald, and J. D. Quigley. “The 2-primary Hurewicz image of tmf”. English. Geom. Topol.27.7 (2023), pp. 2763–2831.issn: 1465-3060. doi: 10.2140/gt.2023. 27.2763

    2023 doi

  2. [9]

    On the existence of the self mapυ9 2 on the Smith-Toda complex V (1) at the prime 3

    M. Behrens and S. Pemmaraju. “On the existence of the self mapυ9 2 on the Smith-Toda complex V (1) at the prime 3”. English.Homotopy theory: relations with algebraic geome- try, group cohomology, and algebraicK-theory. Papers from the international conference on algebraic topo...

    2002

  3. [10]

    A class of2-local finite spectra which admit av1 2-self-map

    P. Bhattacharya and P. Egger. “A class of2-local finite spectra which admit av1 2-self-map”. Advances in Mathematics360 (2020)

    2020

  4. [11]

    Multiplicative structures on Moore spectra.Preprint,arXiv:2203.14787[math.AT] (2022)

    R.Burklund. Multiplicative structures on Moore spectra.Preprint,arXiv:2203.14787[math.AT] (2022). 2022

    2022 arXiv

  5. [13]

    Cofreeness in real bordism theory and the Segal conjecture

    C. Carrick. “Cofreeness in real bordism theory and the Segal conjecture”.Proceedings of the American Mathematical Society150.7 (2021), pp. 3161–3175

    2021

  6. [14]

    On M U-homology of connective models of higher real K- theories

    C. Carrick and M. A. Hill. “On M U-homology of connective models of higher real K- theories”. Proceedings of the American Mathematical Society(2025). To appear

    2025

  7. [15]

    The homological slice spectral sequence in motivic and Real bordism

    C. Carrick, M. A. Hill, and D. C. Ravenel. “The homological slice spectral sequence in motivic and Real bordism”.Advances in Mathematics458 (2024), p. 109955. issn: 0001- 8708

    2024

  8. [16]

    The action of the Morava stabilizer group on the Lubin- Tate moduli space of lifts

    E. S. Devinatz and M. J. Hopkins. “The action of the Morava stabilizer group on the Lubin- Tate moduli space of lifts”.Amer. J. Math.117.3 (1995), pp. 669–710.issn: 0002-9327.doi: 10.2307/2375086

    1995 doi

  9. [17]

    Nilpotence and Stable Homotopy Theory I

    E. S. Devinatz, M. J. Hopkins, and J. H. Smith. “Nilpotence and Stable Homotopy Theory I”. Annals of Mathematics128.2 (1988), pp. 207–241.issn: 0003486X

    1988

  10. [18]

    Cobordism theory with reality

    M. Fujii. “Cobordism theory with reality”. Math J. Okayama Univ.18.2 (1975), pp. 171– 188. 28 REFERENCES

    1975

  11. [19]

    A resolution of the K(2)-local sphere at the prime 3

    P. G. Goerss, H.-W. Henn, M. Mahowald, and C. Rezk. “A resolution of the K(2)-local sphere at the prime 3”. English.Ann. Math. (2) 162.2 (2005), pp. 777–822. issn: 0003- 486X. doi: 10.4007/annals.2005.162.777

    2005 doi

  12. [20]

    Gorenstein duality for real spectra

    J. P. C. Greenlees and L. Meier. “Gorenstein duality for real spectra”.Algebraic and Geo- metric Topology17.6 (), pp. 3547–3619

  13. [21]

    RedshiftandmultiplicationfortruncatedBrown–Petersonspectra

    J.HahnandD.Wilson.“RedshiftandmultiplicationfortruncatedBrown–Petersonspectra”. Annals of Mathematics196.3 (2022), pp. 1277–1351.doi: 10.4007/annals.2022.196.3.6

    2022 doi

  14. [22]

    Finite Subgroups of Division Algebras over Local Fields

    T. Hewett. “Finite Subgroups of Division Algebras over Local Fields”.Journal of Algebra 173.3 (1995), pp. 518–548.issn: 0021-8693. doi: https://doi.org/10.1006/jabr.1995. 1101

    1995 doi

  15. [23]

    On the non-existence of elements of Kervaire invariant one

    M. A. Hill, M. J. Hopkins, and D. C. Ravenel. “On the non-existence of elements of Kervaire invariant one”.Annals of Mathematics184.1 (2016), pp. 1–262

    2016

  16. [24]

    M. A. Hill, M. J. Hopkins, and D. C. Ravenel.Equivariant stable homotopy theory and the Kervaire invariant problem. Vol. 40. New Mathematical Monographs. Cambridge University Press, Cambridge, 2021, pp. ix+870.isbn: 978-1-108-83144-4.doi: 10.1017/9781108917278

    2021 doi

  17. [25]

    The slice spectral sequence for theC4 analog of real K-theory

    M. A. Hill, M. J. Hopkins, and D. C. Ravenel. “The slice spectral sequence for theC4 analog of real K-theory”. Forum Math.29 (2 2017), pp. 383–447

    2017

  18. [26]

    The C2-spectrum Tmf 1(3) and its invertible modules

    M. A. Hill and L. Meier. “The C2-spectrum Tmf 1(3) and its invertible modules”.Algebr. Geom. Topol.17.4 (2017), pp. 1953–2011.issn: 1472-2747.doi: 10.2140/agt.2017.17.1953

    2017 doi

  19. [27]

    The slice spectral sequence of aC4-equivariant height-4 Lubin-Tate theory

    M. A. Hill, X. D. Shi, G. Wang, and Z. Xu. “The slice spectral sequence of aC4-equivariant height-4 Lubin-Tate theory”.Mem. Amer. Math. Soc.288.1429 (2023), pp. v+119. issn: 0065-9266,1947-6221. doi: 10.1090/memo/1429

    2023 doi

  20. [28]

    Nilpotence and stable homotopy theory. II

    M. J. Hopkins and J. H. Smith. “Nilpotence and stable homotopy theory. II.” English.Ann. Math. (2)148.1 (1998), pp. 1–49.issn: 0003-486X; 1939-8980/e

    1998

  21. [29]

    Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence

    P. Hu and I. Kriz. “Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence”.Topology40.2 (2001), pp. 317–399

    2001

  22. [30]

    Stable homotopy groups of spheres

    D. C. Isaksen, G. Wang, and Z. Xu. “Stable homotopy groups of spheres”.Proceedings of the National Academy of Sciences117.40 (2020), pp. 24757–24763. doi: 10 . 1073 / pnas . 2012335117. eprint: https://www.pnas.org/doi/pdf/10.1073/pnas.2012335117

    2020 doi

  23. [31]

    Conjugations on complex manifolds and equivariant homotopy ofM U

    P. S. Landweber. “Conjugations on complex manifolds and equivariant homotopy ofM U”. Bull. Amer. Math. Soc.74 (1968), pp. 271–274

    1968

  24. [32]

    D. J. Lee and P. Pstrągowski. The monochromatic Hahn-Wilson conjecture. 2024. arXiv: 2410.08029 [math.AT]

    2024 arXiv

  25. [33]

    I. Levy. The algebraic K-theory of the K(1)-local sphere via TC. 2022. arXiv:2209.05314 [math.AT]

    2022 arXiv

  26. [34]

    The Image of J in the EHP Sequence

    M. Mahowald. “The Image of J in the EHP Sequence”.Annals of Mathematics116.1 (1982), pp. 65–112. issn: 0003486X, 19398980

    1982

  27. [35]

    Brown-Comenetz duality and the Adams spectral sequence

    M. Mahowald and C. Rezk. “Brown-Comenetz duality and the Adams spectral sequence”. Amer. J. Math184 (1999), pp. 1153–1177

    1999

  28. [36]

    Meier, X

    L. Meier, X. D. Shi, and M. Zeng. Transchromatic phenomena in the equivariant slice spectral sequence. 2024. arXiv:2403.00741 [math.AT]

    2024 arXiv

  29. [37]

    Periodic phenomena in the Adams-Novikov spectral sequence

    H. R. Miller, D. C. Ravenel, and W. S. Wilson. “Periodic phenomena in the Adams-Novikov spectral sequence.” English. Ann. Math. (2) 106 (1977), pp. 469–516. issn: 0003-486X; 1939-8980/e

    1977

  30. [38]

    Nardin and J

    D. Nardin and J. Shah. Parametrized and equivariant higher algebra. 2022. arXiv: 2203. 00072 [math.AT]

    2022

  31. [39]

    The Smith-Toda complex V ((p + 1)/2) does not exist

    L. S. Nave. “The Smith-Toda complex V ((p + 1)/2) does not exist”.Ann. of Math.171.1 (2010), pp. 491–509

    2010

  32. [40]

    On the formal group laws of unoriented and complex cobordism theory

    D. Quillen. “On the formal group laws of unoriented and complex cobordism theory”.Bull. Amer. Math. Soc.75 (1969), pp. 1293–1298.issn: 0002-9904. doi: 10.1090/S0002- 9904- 1969-12401-8

    1969 doi

  33. [41]

    Notes on the Hopkins-Miller theorem

    C. Rezk. “Notes on the Hopkins-Miller theorem”.Homotopy theory via algebraic geometry and group representations (Evanston, IL, 1997). Vol. 220. Contemp. Math. Amer. Math. Soc., Providence, RI, 1998, pp. 313–366.doi: 10.1090/conm/220/03107. REFERENCES 29 Mathematical Institute,...

    1997 doi

  34. [386]

    issn: 0033-5606; 1464-3847/e

  35. [9383]

    doi: 10.1016/0040-9383(66)90004-8

    doi

Pith tools

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