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Computations in Equivariant Topological Hochschild Homology

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Pith's one-line read For odd primes, the C_p-equivariant topological Hochschild homology of HF_p is a specific subring of a divided power algebra, showing equivariant periodicity is richer than the classical one.

desk verdict New odd-prime ETHH computations that look correct; the only real risk is the external E2-formality input behind the geometric fixed points, worth a referee's attention. read the letter →

arxiv 2608.11376 v1 pith:ZYIYBU5M submitted 2026-08-11 math.AT math.KT

David Chan , Marc Gotliboym , Inbar Klang , Noah Wisdom This is my paper · ORCID
classification math.ATmath.KT MSC 55P9155N2219D55
keywords equivarianttopologicalHochschildhomologyTatesquaregeometricfixedpointsThomspectracomplexcobordismdividedpoweralgebrastracemethodsalgebraicK-theory
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 computes the equivariant topological Hochschild homology (ETHH) of two families of basic equivariant ring spectra: Eilenberg–MacLane spectra for F_p with a cyclic group action, for odd primes, and equivariant complex cobordism spectra. For the F_p case, the central result is a complete description of the C_p-fixed homotopy ring as a specific subring of a divided power algebra, a direct equivariant analogue of the classical periodicity theorem for THH(HF_p). The computation shows that the equivariant version of periodicity is substantially richer than the classical one: the answer involves divided powers and an exterior generator, not just a single polynomial generator. For cobordism spectra, the paper identifies ETHH(MU_G) and ETHH(MU_R) as extended Thom spectra and computes their homotopy rings. These computations are intended as input for future work on equivariant topological cyclic homology and trace methods for equivariant algebraic K-theory.

What carries the argument

Three mechanisms carry the computations. The Tate square, a homotopy pullback relating the fixed points, geometric fixed points, homotopy fixed points, and Tate fixed points of a C_p-spectrum, is used to reconstruct \pi_*^{C_p}(\mathrm{ETHH}(H\mathbb{F}_p)) from the geometric and Tate sides. The geometric-fixed-point side is tamed by two facts: geometric fixed points commute with ETHH, and the geometric fixed points of H\mathbb{F}_p are intrinsically formal as an E2-DGA, so they decompose as H\mathbb{F}_p\wedge $S^{1}$_+\wedge (\$\Omega$ $S^{3}$)_+. For the Thom-spectrum results, the central object is equivariant factorization homology together with the equivariant Thom spectrum functor, which promotes the identification of ETHH of a Thom spectrum to an equivalence of E_\infty ring spectra; a multiplicative equivariant bar spectral sequence and an equivariant cellular decomposition of the infinite special unitary group then yield the homotopy rings.

What would settle it

Construct an E2-DGA over \mathbb{F}_p with homology \mathbb{F}_p[y], |y|=2, that is not quasi-isomorphic to the formal algebra \mathbb{F}_p[y]; its existence would invalidate Proposition 5.11 and with it the geometric-fixed-point computations of Theorems A and B.

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Extended reading notes

Core claim

The paper's central discovery is a complete computation of the C_p-fixed homotopy ring of equivariant topological Hochschild homology of the Eilenberg–MacLane spectrum H\mathbb{F}_p, for odd primes p. Writing \Gamma for divided powers and \Lambda for exterior algebras, the answer is the subring of \Gamma[a]\otimes \mathbb{F}_p[b,c]\otimes \Lambda[d,e] generated by b and by all monomials divisible by either e or a divided power of a, with a,b,c in degree 2, d in degree 1, and e in degree 3. In the non-equivariant limit this reduces to the classical periodicity theorem \pi_*\mathrm{THH}(H\mathbb{F}_p)\cong \mathbb{F}_p[b], so the theorem shows that equivariant periodicity is strictly more intricate. The method is a Tate-square analysis: the geometric fixed points are computed first, as H\mathbb{F}_p\wedge (\$\Omega$ $S^{3}$)_+\wedge $S^{1}$_+\wedge CP^\infty_+\wedge ($LS^{3}$)_+, giving the full ring, and the fixed-point subring is then isolated by tracking the map from geometric to Tate fixed points on generators. For equivariant complex cobordism, the paper identifies \mathrm{ETHH}(MU_G) with MU_G\wedge \Sigma^\infty_+ SU_G and \mathrm{ETHH}(MU_\mathbb{R}) with MU_\mathbb{R}\wedge \Sigma^\infty_+ B(BU^\mathbb{R}), with explicit homotopy rings.

Load-bearing premise

The load-bearing premise is that the graded algebra \mathbb{F}_p[y] in degree 2 is the only E2-DGA with that homology; if that cited formality statement fails, the geometric fixed point decomposition behind Theorems A and B collapses.

Editorial extensions

If this is right

  • If Theorem A is right, any future computation of equivariant topological cyclic homology for H\mathbb{F}_p must absorb a divided-power and exterior structure rather than a single polynomial generator, so the equivariant trace-method pipeline will be qualitatively more involved than the classical one.
  • Theorem B provides a concrete topological model for the geometric fixed points, H\mathbb{F}_p\wedge (\Omega S^3)_+\wedge S^1_+\wedge CP^\infty_+\wedge (LS^3)_+, which can serve as a starting point for RO(C_p)-graded computations of the same spectrum.
  • The identification \mathrm{ETHH}(MU_G)\simeq MU_G\wedge \Sigma^\infty_+SU_G for finite G means the homotopy groups of equivariant complex cobordism's ETHH are determined by the equivariant homology of SU_G; for finite abelian G, the paper computes these as exterior algebras over the equivariant cobordism rings of subgroups.
  • For the Real cobordism spectrum MU_\mathbb{R}, the RO(C_2)-graded homotopy is an exterior algebra over (MU_\mathbb{R})_\star on classes of degree n\rho+1, giving a complete description of \mathrm{ETHH}(MU_\mathbb{R}).
  • When the characteristic is prime to the group order, \pi^G_*\mathrm{ETHH}(H\mathbb{F})\cong \pi_*\mathrm{THH}(H\mathbb{F}), so the exotic behavior of Theorem A is specific to the modular case where the prime divides |G|.

Reading between the lines

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

  • The paper leaves the RO(C_p)-graded ring of \mathrm{ETHH}(H\mathbb{F}_p) open; one plausible extension is that divided powers become polynomial in that grading, converting the subring of Theorem A into a cleaner free-algebra description.
  • The same Tate-square strategy should work for H\mathbb{F}_p with larger groups G, but only where the geometric fixed points admit an intrinsic formality decomposition; the non-modular theorem suggests the interesting cases are exactly the primes dividing |G|.
  • The multiplicative equivariant bar spectral sequence developed here could be applied to other equivariant Thom spectra of E_\infty maps, producing RO(G)-graded Green-functor computations beyond the two examples in the paper.
  • A direct test of the paper's framework would be to adapt its method to p=2; since the geometric fixed points of H\mathbb{F}_2 have a different form, the resulting C_2-fixed ring is expected to differ qualitatively from the odd-prime answer.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper computes equivariant topological Hochschild homology (ETHH) for several fundamental ring G-spectra. The main results are: for odd p, a complete description of the C_p-graded homotopy ring of ETHH(HF_p) (Theorem A), obtained via the Tate square together with a new computation of the C_p-geometric fixed points (Theorem B); and computations of ETHH for the equivariant complex cobordism spectra MU_G and MU_R (Theorems C–E). The proofs use the Tate square, geometric fixed points, equivariant factorization homology and Thom spectra, a new multiplicative equivariant bar spectral sequence, and explicit G-CW structures. The paper is computational in character and explicitly positions the results as input for future work on equivariant trace methods and equivariant topological cyclic homology.

Significance. If the computations are correct, this is a substantial contribution to a young area: it supplies the first nontrivial odd-primary equivariant Bokstedt periodicity computation and gives explicit multiplicative descriptions of ETHH for the standard equivariant cobordism spectra. The paper also develops tools of independent interest, especially the multiplicative G-equivariant bar spectral sequence and the upgrade of the equivariant Thom spectrum/factorization homology compatibility to multiplicative statements. A clear strength is that the central computations are checked against external benchmarks: Bokstedt's non-equivariant computation of THH(HF_p), the established computation of THH(MU), and the already-existing equivariant factorization homology framework. The reliance on the authors' own and collaborators' prior work is not circular, since the cited results are independent of the present computations.

major comments (3)
  1. [§5.1, Proposition 5.11] This proposition is the only bridge between the geometric fixed points of ETHH(HF_p) and a smash product whose THH is computable, so Theorems A and B rest on it. The proof asserts an equivalence of E_2-HF_p-algebras HF_p^{ΦC_p} ≃ HF_p ∧ S^1_+ ∧ (ΩS^3)_+, but the argument as written does not supply the required E_2-algebra maps. For the F_p[x]/x^2 factor, the text invokes only CDGA intrinsic formality, 'straightforwardly checked from the definition,' and for F_p[y] it cites E_2 formality from [Hor25, Theorem 3.5] and [BM22, Theorem 2.1]. It is not explained why these formality statements can be upgraded to maps of E_2-HF_p-algebras with the specified effect on generators, nor why the two chosen equivalences are compatible under the smash product. The surjectivity and finite-dimensionality argument is fine once such an E_2 map exists, but the existence of the map is exactly the load-bearing point. Please provide a full proof or a direct reference that covers the E_2-algebra statement, not merely the underlying DGA or CDGA formality.
  2. [§4.2, proof of Theorem 4.11] The base case of the induction is incorrect as stated: SUG(C) is SU(1), which is the trivial group, not S^1. If a degree-1 exterior generator were introduced, it would contradict the theorem's own degrees, where the first exterior generator λ_1 has degree 2·1+1 = 3. The induction should begin with n = 2, where SUG(W_2) ≃ SU(2) ≃ S^3, or the proof should explicitly use U(1) if that is the intended object. Since Theorem D and Corollary 4.12 depend on this proof, the base case needs to be fixed.
  3. [§5.1, Lemma 5.14] The proof of the computation of H_*(LS^3; F_p) is not correct as written. In the Serre spectral sequence for ΩS^3 → LS^3 → S^3, the generator of H^*(S^3) has bidegree (3,0) and the polynomial generator of H^*(ΩS^3) has bidegree (0,2), so d_2 of the fiber generator is not a possible target; the displayed 'd_2(e) = kd' is also dimensionally inconsistent. The argument also writes LS^2 for LS^3. The statement itself is standard, and the collapse can be justified either by degree reasons or by the product splitting LS^3 ≃ S^3 × ΩS^3, but the proof should be rewritten. Since Corollary 5.15 and hence Theorem B use this computation, the correction is not purely cosmetic.
minor comments (4)
  1. [§5.3, Proposition 5.18] The E_∞ page is displayed as Λ(x) ⊗ F_p[x,y,b], which is inconsistent: x cannot be both an exterior and a polynomial generator. It should presumably read Λ(x) ⊗ F_p[y^{±1}, b], matching Proposition 5.19.
  2. [§5.4, Proposition 5.23] The last bullet says 'where each z_i is in the image of j_i,' but no elements z_i have been defined. Please either define them or delete the phrase.
  3. [§5.1, Lemma 5.14] In the same lemma, the notation is inconsistent: the polynomial generator in degree 2 is called c in the statement but d in the proof, and the phrase 'determined by is d_2(e)' contains a grammatical error.
  4. [§5.4, Theorem 5.29 and Theorem A] The description 'generated by b and all monomials divisible by either e or a divided power of a' should clarify that the divided powers are the positive ones γ_n(a), n ≥ 1, so that the reader does not read γ_0(a) = 1 as making the generating set trivial.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorems A and B reduce to external formality results and independent spectral sequence computations, not to their own conclusions.

full rationale

The central derivation chain is self-contained in the required sense. Theorem A is assembled from the Tate square, the homotopy fixed point and Tate spectral sequences, and explicit computations of the map q_* on generators; none of these steps is a fitted parameter renamed as a prediction or an equation identical to the statement being proved. The geometric fixed point identification of Corollary 5.7 follows from symmetric monoidality of geometric fixed points, and Proposition 5.11 is the only bridge to a computable smash product; its proof rests on intrinsic formality of F_p[y] and F_p[x]/x^2 cited to Hor25 and BM22, which are works by non-authors. The non-equivariant benchmark THH(HF_p) ≃ HF_p ∧ (ΩS^3)_+ is likewise cited externally. The paper does cite prior work by its own authors, namely CGK25 for the definition of ETHH, HHK+24 for equivariant Thom/factorization homology technology, Wis25 for a cellular lemma, and CV25 for a Tate-square lemma; these are prior results with proofs and are not simply renamed versions of Theorem A or B. If the E2-formality assertions in Proposition 5.11 fail, the geometric fixed point computation would collapse, but that is an external correctness risk and not a circular reduction. No step was found in which the paper's input equals its claimed output by construction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The computations rest on standard equivariant stable homotopy results and on three external pillars: Bokstedt's THH(HF_p), intrinsic formality results, and the HHK+24 Thom spectrum/factorization homology machinery. No free parameters or invented entities are introduced; the paper's contribution is the derivation itself.

assumptions (6)
  • standard math The Tate square is a homotopy pullback of ring spectra for genuine G-spectra.
    Used throughout Section 5 to compute fixed points from geometric, homotopy fixed, and Tate constructions.
  • standard math Geometric fixed points form a symmetric monoidal left adjoint.
    Invoked in Corollary 5.7 to commute ETHH with ΦC_p.
  • domain assumption Bokstedt's computation π_*THH(HF_p)≅F_p[b] with |b|=2.
    Used as the external baseline in Section 5; the equivariant computation is compared against it.
  • domain assumption Intrinsic formality of the F_p-DGAs F_p[y] and F_p[x]/x^2 as cited from [Hor25, Theorem 3.5] and [BM22, Theorem 2.1].
    Proposition 5.11 uses this to identify HF_p^{ΦC_p} as an E2-HF_p-algebra; it is the weakest external premise.
  • domain assumption Equivariant Thom spectra commute with equivariant factorization homology, and factorization homology of ΩVX is identified with mapping spaces (HHK+24, Theorems 3 and 4).
    Used in Corollary 4.3 and Theorem 4.2 to compute ETHH of Thom spectra.
  • domain assumption Cole's computation of MUG-homology of Σ^1CP(W_n) and the equivariant Künneth theorem [LM06].
    Used in Theorem 4.11 to show attaching maps induce zero in MUG-homology.

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Pith. "Pith review of Computations in Equivariant Topological Hochschild Homology." pith.science (2026). https://pith.science/paper/ZYIYBU5M

@misc{pith2026260811376,
  author       = {Pith},
  title        = {Pith review of: Computations in Equivariant Topological Hochschild Homology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZYIYBU5M}},
  note         = {Machine review of arXiv:2608.11376}
}
abstract

One of the most effective approaches to computations in algebraic $K$-theory is trace methods, which compare algebraic $K$-theory with topological Hochschild homology and topological cyclic homology. In recent work, two of the authors, together with Gerhardt, construct an equivariant refinement of topological Hochschild homology ($\mathrm{ETHH}$) which receives a trace map from Merling's genuine equivariant algebraic $K$-theory. In this paper, we perform foundational computations of $\mathrm{ETHH}$ that can serve as input for future computations of $\mathrm{ETHH}$ and equivariant topological cyclic homology. Namely, we compute $\mathrm{ETHH}(H\underline{\mathbb{F}}_p)$ for odd primes, showcasing the complexity of B\"okstedt periodicity in this setting. Furthermore, we give computations of $\mathrm{ETHH}$ for the equivariant complex cobordism spectra $MU_G$ and $MU_{\mathbb{R}}$.

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Works this paper leans on

67 extracted references · 40 canonical work pages

  1. [1]

    Topology Appl

    Adamyk, Katharine and Gerhardt, Teena and Hess, Kathryn and Klang, Inbar and Kong, Hana Jia , TITLE =. Topology Appl. , FJOURNAL =. 2022 , PAGES =. doi:10.1016/j.topol.2022.108102 , URL =

    arXiv 2022

  2. [2]

    Bhatt, Bhargav and Morrow, Matthew and Scholze, Peter , TITLE =. Publ. Math. Inst. Hautes \'Etudes Sci. , FJOURNAL =. 2019 , PAGES =. doi:10.1007/s10240-019-00106-9 , URL =

    doi:10.1007/s10240-019-00106-9 2019

  3. [3]

    Behrens, Mark and Wilson, Dylan , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 2018 , NUMBER =. doi:10.1090/proc/14175 , URL =

    doi:10.1090/proc/14175 2018

  4. [4]

    and Cohen, Ralph L

    Blumberg, Andrew J. and Cohen, Ralph L. and Schlichtkrull, Christian , TITLE =. Geom. Topol. , FJOURNAL =. 2010 , NUMBER =. doi:10.2140/gt.2010.14.1165 , URL =

    doi:10.2140/gt.2010.14.1165 2010

  5. [5]

    Trace methods for equivariant algebraic K-theory

    David Chan and Teena Gerhardt and Inbar Klang , year=. Trace methods for equivariant algebraic. 2505.11327 , archivePrefix=

    work page Pith review arXiv

  6. [6]

    Greenlees, J. P. C. and May, J. P. , TITLE =. Mem. Amer. Math. Soc. , FJOURNAL =. 1995 , NUMBER =. doi:10.1090/memo/0543 , URL =

    doi:10.1090/memo/0543 1995

  7. [7]

    Equivariant nonabelian

    Jeremy Hahn and Asaf Horev and Inbar Klang and Dylan Wilson and Foling Zou , year=. Equivariant nonabelian. 2006.13348 , archivePrefix=

    arXiv 2006

  8. [8]

    Geometry & Topology , number =

    Jeremy Hahn and Dylan Wilson , title =. Geometry & Topology , number =. 2020 , doi =

    work page 2020

Show all 67 references
  1. [9]

    Hahn, Jeremy and Wilson, Dylan , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2022 , NUMBER =. doi:10.4007/annals.2022.196.3.6 , URL =

    2022 doi

  2. [10]

    Hill, M. A. and Hopkins, M. J. and Ravenel, D. C. , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2016 , NUMBER =. doi:10.4007/annals.2016.184.1.1 , URL =

    2016 doi

  3. [11]

    Mathematische Zeitschrift , volume=

    Strictly commutative complex orientation theory , author=. Mathematische Zeitschrift , volume=. 2018 , publisher=

    2018

  4. [12]

    2019 , eprint=

    Genuine equivariant factorization homology , author=. 2019 , eprint=

    2019

  5. [13]

    Krause, Achim and Nikolaus, Thomas , TITLE =. Compos. Math. , FJOURNAL =. 2022 , NUMBER =. doi:10.1112/s0010437x22007655 , URL =

    2022 doi

  6. [14]

    , TITLE =

    Kronholm, William C. , TITLE =. Homology Homotopy Appl. , FJOURNAL =. 2010 , NUMBER =. doi:10.4310/hha.2010.v12.n1.a7 , URL =

    2010 doi

  7. [15]

    Lashof, R. K. and May, J. P. and Segal, G. B. , TITLE =. Proceedings of the. 1983 , ISBN =. doi:10.1090/conm/019/711050 , URL =

    1983 doi

  8. [16]

    Equivariant universal coefficient and

    Lewis, L Gaunce and Mandell, Michael A , journal=. Equivariant universal coefficient and. 2006 , publisher=

    2006

  9. [17]

    Mehrle, David and Quigley, J. D. and Stahlhauer, Michael , title =. Bulletin of the London Mathematical Society , volume =. doi:https://doi.org/10.1112/blms.70138 , url =

    doi

  10. [18]

    The algebraic

    David Chan and Noah Wisdom , year=. The algebraic. 2508.14207 , archivePrefix=

    arXiv

  11. [19]

    Noah Wisdom , year=. The. 2604.09954 , archivePrefix=

    arXiv

  12. [20]

    2405.03578 , archivePrefix=

    Elden Elmanto and Ningchuan Zhang , year=. 2405.03578 , archivePrefix=

    arXiv

  13. [21]

    and McCarthy, Randy , TITLE =

    Dundas, Bj rn Ian and Goodwillie, Thomas G. and McCarthy, Randy , TITLE =. 2013 , PAGES =

    2013

  14. [22]

    Hesselholt, Lars and Madsen, Ib , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2003 , NUMBER =. doi:10.4007/annals.2003.158.1 , URL =

    2003 doi

  15. [23]

    Derived induction theory for the

    Chase Vogeli , year=. Derived induction theory for the. 2510.25763 , archivePrefix=

  16. [24]

    Equivariant localizing motives and multiplicative norms on algebraic

    Kaif Hilman and Maxime Ramzi , year=. Equivariant localizing motives and multiplicative norms on algebraic. 2603.15873 , archivePrefix=

  17. [25]

    Malkiewich, Cary and Merling, Mona , TITLE =. Doc. Math. , FJOURNAL =. 2019 , PAGES =

    2019

  18. [26]

    Malkiewich, Cary and Merling, Mona , TITLE =. Adv. Math. , FJOURNAL =. 2022 , PAGES =. doi:10.1016/j.aim.2022.108242 , URL =

    2022

  19. [27]

    and Hsiang, W

    B\"okstedt, M. and Hsiang, W. C. and Madsen, I. , TITLE =. Invent. Math. , FJOURNAL =. 1993 , NUMBER =. doi:10.1007/BF01231296 , URL =

    1993 doi

  20. [28]

    2024 , eprint=

    Koszul Resolutions over Free Incomplete Tambara Functors for Cyclic p -Groups , author=. 2024 , eprint=

    2024

  21. [29]

    Lecture notes in mathematics , volume=

    Equivariant stable homotopy theory , author=. Lecture notes in mathematics , volume=. 1986 , PAGES =. doi:10.1007/BFb0075778 , URL =

    1986 doi

  22. [30]

    Linskens, Sil and Nardin, Denis and Pol, Luca , TITLE =. Geom. Topol. , FJOURNAL =. 2025 , NUMBER =. doi:10.2140/gt.2025.29.1345 , URL =

    2025 doi

  23. [31]

    and May, J

    Guillou, Bertrand J. and May, J. Peter and Merling, Mona , TITLE =. Algebr. Geom. Topol. , FJOURNAL =. 2017 , NUMBER =. doi:10.2140/agt.2017.17.2565 , URL =

    2017 doi

  24. [32]

    2017 , journal=

    Higher algebra , author=. 2017 , journal=

    2017

  25. [33]

    Marzantowicz, Wac aw and Prieto, Carlos , TITLE =. J. London Math. Soc. (2) , FJOURNAL =. 2004 , NUMBER =. doi:10.1112/S0024610703004721 , URL =

    2004 doi

  26. [34]

    McClure, J. E. and Staffeldt, R. E. , TITLE =. Amer. J. Math. , FJOURNAL =. 1993 , NUMBER =. doi:10.2307/2374721 , URL =

    1993 doi

  27. [35]

    Merling, Mona , TITLE =. Math. Z. , FJOURNAL =. 2017 , NUMBER =. doi:10.1007/s00209-016-1745-3 , URL =

    2017 doi

  28. [36]

    Quigley, J. D. and Shah, Jay , TITLE =. J. Topol. , FJOURNAL =. 2025 , NUMBER =. doi:10.1112/topo.70018 , URL =

    2025 doi

  29. [37]

    2022 , eprint=

    On the equivalence of two theories of real cyclotomic spectra , author=. 2022 , eprint=

    2022

  30. [38]

    Rognes, John , TITLE =. J. Topol. , FJOURNAL =. 2020 , NUMBER =. doi:10.1112/topo.12141 , URL =

    2020 doi

  31. [39]

    Journal of the Institute of Polytechnics, Osaka City University

    Ichiro Yokota , title =. Journal of the Institute of Polytechnics, Osaka City University. Series A: Mathematics , number =

  32. [40]

    2001 , PAGES =

    McCleary, John , TITLE =. 2001 , PAGES =

    2001

  33. [41]

    and Wilson, W

    Ravenel, Douglas C. and Wilson, W. Stephen , TITLE =. Amer. J. Math. , FJOURNAL =. 1980 , NUMBER =. doi:10.2307/2374093 , URL =

    1980 doi

  34. [42]

    Algebraic topology , SERIES =

    Hesselholt, Lars , TITLE =. Algebraic topology , SERIES =. 2009 , ISBN =. doi:10.1007/978-3-642-01200-6\_7 , URL =

    2009 doi

  35. [43]

    Greenlees, J. P. C. and May, J. P. , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1997 , NUMBER =. doi:10.2307/2952455 , URL =

    1997 doi

  36. [44]

    Hausmann, Markus , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2022 , NUMBER =. doi:10.4007/annals.2022.195.3.2 , URL =

    2022 doi

  37. [45]

    Araki, Sh\^or\^o , TITLE =. Japan. J. Math. (N.S.) , FJOURNAL =. 1979 , NUMBER =. doi:10.4099/math1924.5.403 , URL =

    1979 doi

  38. [46]

    , TITLE =

    Hill, Michael A. , TITLE =. J. Topol. , FJOURNAL =. 2022 , NUMBER =. doi:10.1112/topo.12227 , URL =

    2022 doi

  39. [47]

    Ausoni, Christian , TITLE =. Amer. J. Math. , FJOURNAL =. 2005 , NUMBER =

    2005

  40. [48]

    , TITLE =

    Hunter, Thomas J. , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 1996 , NUMBER =. doi:10.1090/S0002-9947-96-01742-4 , URL =

    1996 doi

  41. [49]

    Elmendorf, A. D. and Kriz, I. and Mandell, M. A. and May, J. P. , TITLE =. 1997 , PAGES =. doi:10.1090/surv/047 , URL =

    1997 doi

  42. [50]

    Angeltveit, Vigleik and Rognes, John , TITLE =. Algebr. Geom. Topol. , FJOURNAL =. 2005 , PAGES =. doi:10.2140/agt.2005.5.1223 , URL =

    2005 doi

  43. [51]

    Bruner, R. R. and May, J. P. and McClure, J. E. and Steinberger, M. , TITLE =. 1986 , PAGES =. doi:10.1007/BFb0075405 , URL =

    1986 doi

  44. [52]

    Horel, Geoffroy , TITLE =. Ann. H. Lebesgue , FJOURNAL =. 2025 , PAGES =. doi:10.5802/ahl.246 , URL =

    2025 doi

  45. [53]

    Algebraic. Trans. Amer. Math. Soc. , FJOURNAL =. 2022 , NUMBER =. doi:10.1090/tran/8613 , URL =

    2022 doi

  46. [54]

    2002 , PAGES =

    Hatcher, Allen , TITLE =. 2002 , PAGES =

    2002

  47. [55]

    A general

    Serge Bouc and Ivo Dell'Ambrogio and Rubén Martos , year=. A general. 2405.18885 , archivePrefix=

    arXiv

  48. [56]

    Equivariant Formal Group Laws , volume =

    Michael Cole and John Greenlees and Igor Kriz , year =. Equivariant Formal Group Laws , volume =. Proceedings of the London Mathematical Society , doi =

  49. [57]

    Wisdom, Noah , TITLE =. Trans. Amer. Math. Soc. Ser. B , FJOURNAL =. 2025 , PAGES =. doi:10.1090/btran/224 , URL =

    2025 doi

  50. [58]

    1996 , booktitle=

    Calculations in complex equivariant bordism , author=. 1996 , booktitle=

    1996

  51. [59]

    1973 , journal=

    Equivariant unitary cobordism and classifying spaces , author=. 1973 , journal=

    1973

  52. [60]

    2018 , publisher=

    Global homotopy theory , author=. 2018 , publisher=

    2018

  53. [61]

    Schlank , year=

    Robert Burklund and Jeremy Hahn and Ishan Levy and Tomer M. Schlank , year=. 2310.17459 , archivePrefix=

    arXiv

  54. [62]

    Mathematische Zeitschrift , year=

    The universality of equivariant complex bordism , author=. Mathematische Zeitschrift , year=

  55. [63]

    Equivariant Topological Cyclic Homology , author=

  56. [64]

    Chan, David and Vogeli, Chase , TITLE =. Proc. Lond. Math. Soc. (3) , FJOURNAL =. 2025 , NUMBER =. doi:10.1112/plms.70012 , URL =

    2025 doi

  57. [65]

    and Mehrle, David and Quigley, James D

    Hill, Michael A. and Mehrle, David and Quigley, James D. , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2023 , NUMBER =. doi:10.1093/imrn/rnab361 , URL =

    2023 doi

  58. [66]

    Barwick, Clark , TITLE =. Adv. Math. , FJOURNAL =. 2017 , PAGES =. doi:10.1016/j.aim.2016.08.043 , URL =

    2017 doi

  59. [67]

    Barwick, Clark and Glasman, Saul and Shah, Jay , TITLE =. Tunis. J. Math. , FJOURNAL =. 2020 , NUMBER =. doi:10.2140/tunis.2020.2.97 , URL =

    2020 doi

Pith tools

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