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C_2-equivariant stable homotopy from real motivic stable homotopy

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Pith's one-line read This paper proves that p-complete C2-equivariant stable homotopy is a localization of p-complete cellular real motivic stable homotopy.

desk verdict A genuine structural bridge: p-complete C2-equivariant stable homotopy is a localization of cellular real motivic stable homotopy, with a usable computational recipe; the proof is dense and externally dependent but sound. read the letter →

arxiv 1908.08378 v3 pith:E5NPRTVD submitted 2019-08-22 math.AT math.KT

Mark Behrens , Jay Shah This is my paper · ORCID
classification math.ATmath.KT MSC 55P9114F4255P42
keywords C2-equivariantstablehomotopyrealmotivicBettirealizationlocalizationcellularspectratau-selfmapsisotropyseparationRO(C2)-gradedgroups
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 proves that Betti realization, from real motivic spectra over $\mathbb{R}$ to $C_2$-equivariant spectra, becomes a localization after passing to $p$-complete cellular objects: the right adjoint $\mathrm{Cell}\,\mathrm{Sing}^{C_2}$ embeds the $p$-complete $C_2$-equivariant stable homotopy category fully faithfully into the $p$-complete cellular real motivic stable homotopy category. The point is computational as well as categorical. For a $p$-complete cellular real motivic spectrum $X$, the geometric part of its $C_2$-Betti realization is obtained by inverting the Euler class $\rho$, the homotopy-complete part is obtained by inverting $\tau$ on the $\rho$-completion tower, and the Tate part by inverting both. Because the $C_2$-equivariant homotopy type is recovered from an isotropy separation square built from these three pieces, the $RO(C_2)$-graded equivariant homotopy groups of examples such as $\mathrm{H}\mathbb{F}_2$, $\mathrm{H}\mathbb{Z}_2$, and connective real $K$-theory can be computed from their simpler bigraded motivic homotopy groups.

What carries the argument

The load-bearing comparison map is the non-canonical map $C(\rho) \to \Sigma^\infty_+ \mathrm{Spec}\,\mathbb{C}$ from the cofiber of the real motivic Euler class $\rho$ to the complex variety point; Proposition 8.3 proves it becomes an equivalence after $p$-completion and cellularization, even though $\Sigma^\infty_+ \mathrm{Spec}\,\mathbb{C}$ itself is not cellular. This turns the complex motivic category into modules over $C(\rho)$ inside the real cellular category, so the complex $\tau$-inversion localization can be pulled back to the real side. The remaining input is a supply of $\tau$-self maps on the spectra $C(\rho_i)$: at the prime $2$ these are lifted from $u$-self maps on the equivariant spectra $C(a_i)$, using James periodicity and an isomorphism range between real motivic and $C_2$-equivariant spheres; at odd primes a $\tau^2$-self map exists on every $\rho$-complete real motivic spectrum. Finally, the isotropy separation square for $C_2$-spectra, equivalently the $\rho$-arithmetic square, assembles the geometric, homotopy-complete, and Tate localizations into the full equivariant homotopy type.

What would settle it

For $p=2$, compute the effect of the comparison map $C(\rho)^\wedge_p \to \Sigma_+^\infty \mathrm{Spec}\,\mathbb{C}^\wedge_p$ on mod $2$ motivic homology: Proposition 8.3 predicts exactly the quotient $\mathbb{F}_2[\tau,\rho] \to \mathbb{F}_2[\tau]$, so any class in the kernel beyond $\rho$ would disprove the localization theorem.

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Extended reading notes

Core claim

The central claim is that the adjunction $\widehat{\mathrm{Be}}^{C_2}_p : \mathrm{SH}^{\mathrm{cell}}(\mathbb{R})_p^\wedge \rightleftarrows (\mathrm{Sp}^{C_2})_p^\wedge : \mathrm{Cell}\,\mathrm{Sing}^{C_2}$ is a localization: $\mathrm{Cell}\,\mathrm{Sing}^{C_2}$ is fully faithful. Equivalently, every $p$-complete $C_2$-equivariant spectrum is, up to equivalence, the $p$-complete cellular Betti realization of a $p$-complete cellular real motivic spectrum. On homotopy groups, Theorem 8.26 gives isomorphisms $\pi^{\mathbb{R}}_{*,*}(X^\wedge_\rho[\tau^{-1}]) \cong \pi^{C_2}_{*,*}(\widehat{\mathrm{Be}}^{C_2}_p(X)^h)$ and, via Theorem 8.10, $\pi^{\mathbb{R}}_{*,*}(X[\rho^{-1}]) \cong \pi^{C_2}_{*,*}(\widehat{\mathrm{Be}}^{C_2}_p(X)^\Phi)$. Combining the geometric and homotopy-complete parts with the Tate part through the isotropy separation square recovers the full $RO(C_2)$-graded equivariant homotopy groups.

Load-bearing premise

The whole bridge rests on the claim that the real motivic cofiber of the Euler class $\rho$ can replace the non-cellular spectrum $\Sigma_+^\infty \mathrm{Spec}\,\mathbb{C}$ after $p$-completion and cellularization; if that comparison map fails to be an equivalence, the identification of the complex motivic category with modules over $C(\rho)$ breaks.

Editorial extensions

If this is right

  • For every $p$-complete cellular real motivic spectrum $X$, the $RO(C_2)$-graded homotopy groups of $\widehat{\mathrm{Be}}^{C_2}_p(X)$ are completely determined by the bigraded motivic homotopy groups of $X$ together with the maps in the isotropy separation square.
  • At odd primes the $C_2$-equivariant Tate spectrum of a Betti realization is contractible, so the equivariant homotopy groups split as a direct sum of the $\rho$-inverted and the $\tau^2$-inverted motivic groups.
  • The full faithfulness of $\mathrm{Cell}\,\mathrm{Sing}^{C_2}$ means that constructions and objects in $p$-complete $C_2$-equivariant stable homotopy can be transferred to the real motivic category; the paper uses this to compute the equivariant homotopy of $\mathrm{H}\mathbb{F}_2$, $\mathrm{H}\mathbb{Z}_2$, and $\mathrm{kgl}^\wedge_2$.
  • The $\tau$-self maps give real motivic periodicity: $C(\rho_i)^\wedge_p$ admits a $\tau_j$-self map with $j = 2\gamma(i-1)$ at $p=2$, and every $\rho$-complete real motivic spectrum admits a $\tau^2$-self map at odd primes.

Reading between the lines

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

  • The paper leaves open whether the localization theorem survives without $p$-completion; if it did, the relation between real motivic and $C_2$-equivariant stable homotopy would be a genuine recollement rather than one on completed cellular objects.
  • Because the proof of the $\tau$-self maps at $p=2$ uses only a specific isomorphism range between real motivic and equivariant stems, extending that range would automatically produce $\tau$-self maps in a larger region, making the computational method applicable to more spectra.
  • The negative cone that appears in $\pi^{C_2}_{*,*}\mathrm{H}\mathbb{F}_2$ is presented in the paper as a consequence of local duality; a concrete test would be to see whether the same duality mechanism predicts the negative cones for $\mathrm{H}\mathbb{Z}_2$ and $\mathrm{kR}$ without separate input.
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Editorial analysis

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Referee Report

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Summary. This paper proves that p-complete C2-equivariant stable homotopy is a localization of p-complete cellular real motivic stable homotopy via C2-Betti realization. The main theorem (Theorem 8.22) states that the right adjoint Cell Sing^{C2}: (Sp^{C2})_p^wedge -> SH^{cell}(R)_p^wedge is fully faithful, and Theorem 8.26 gives a computational formula expressing the homotopy groups of the homotopy completion of the C2-Betti realization in terms of motivic homotopy groups of the tau-inverted rho-completion. The proof combines formal recollement lemmas (Section 5), a monoidal Barr-Beck theorem of Mathew-Naumann-Noel, cellularization techniques (Section 4), Bachmann's real etale localization theorem, and explicit tau-self maps on C(rho^i) constructed in Section 7 using the Dugger-Isaksen isomorphism theorem and Stahn's odd-primary results. The paper also works out examples HF_2, HZ_2, and kgl_2^wedge.

Significance. If correct, this establishes a new and powerful bridge between real motivic and C2-equivariant stable homotopy, with direct computational consequences. The formal framework is reusable, and the examples clearly demonstrate the method. The paper is honest about its reliance on external deep results (Bachmann, Heller-Ormsby, Stahn, Dugger-Isaksen, Lin), and the proof chain appears internally coherent. I specifically checked the two fragile points flagged in review: the Dugger-Isaksen range i >= 3j-5 used in Theorem 7.10 is satisfied in the relevant bidegrees, and the eta-completeness assertion in Proposition 8.3 is true, though the written justification for Spec(C)_+ needs a small addition. The authors also give explicit credit to independent work of Isaksen-Kong-Wang-Xu, though the citation is incomplete.

minor comments (4)
  1. [Section 8, Proposition 8.3] The sentence "both C(rho) and Spec C+ are eta-complete by Prop. 6.7" is not literally correct: Prop. 6.7 is stated only for X in SHcell(K)[1/2], and Spec C+ is not cellular (Remark 8.4). Please add a direct argument for the eta-completeness of Spec C+ (for instance, via the vanishing of rho on Spec C+ and the cellular approximation C(rho)), since this is used to justify convergence of the motivic Adams spectral sequence at odd primes.
  2. [Introduction, Theorem 1.7] The sentence "which has also been independently obtained by Isaksen-Kong-Wang-Xu" provides no reference; please add a precise citation or state this as a personal communication, so that readers can locate the independent work.
  3. [Remark 8.17] The phrase "Ricka proves proves this" contains a duplicated "proves", and "metioned" should be "mentioned".
  4. [Section 9] The notation pi^{C2}_* is used both for the Z-graded and the RO(C2)-graded equivariant homotopy groups; please disambiguate, especially in the Mayer-Vietoris arguments that mix bigraded and single-graded inputs.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the localization theorem is assembled from independently established external results and explicit computations, not from its own conclusion.

full rationale

The paper's central claims (Theorem 8.22 full faithfulness of Cell Sing^{C2} and Theorem 8.26 the computation of homotopy completions) are not derived from the target result by definition. Theorem 8.22 is assembled formally: the recollement criteria of Lemma 5.1 are checked using Lemma 8.19, Lemma 8.20, Bachmann's theorem 8.10, and Corollary 8.21; Corollary 8.21 in turn combines Theorem 8.18 (Dugger-Isaksen/Stahn), Corollary 8.6, and the formal lifting result Proposition 3.10. The key calculational input, Proposition 8.3, is proved directly by a motivic HF_p homology computation, not by assuming the localization. The τ-self maps of Section 7 are constructed from C_2-equivariant u-self maps, which are established via Landweber periodicity, Lin's theorem, and BMMS, and are then lifted to the real motivic category using the external Dugger-Isaksen isomorphism theorem (Theorem 7.6); no fitted parameter is renamed as a prediction. The citations to Bachmann, Heller-Ormsby, Stahn, Dugger-Isaksen, Landweber, Lin, and Mathew-Naumann-Noel are citations of independent theorems with stated hypotheses that do not include the present conclusions. The paper also openly flags its own limitations (e.g., Warning 8.7 that Cell is not strong monoidal, Remark 8.17 about the delicate p-completion issue in Ricka's comparison), which further indicates the argument is not being forced by circular self-reference. No step reduces to an input by construction, and there are no load-bearing self-citations.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central claim is built on established external theorems rather than ad hoc assumptions. No free parameters fitted to data and no new postulated entities appear. The most paper-specific input is the existence of tau-self maps on C(rho_i), which is proven in Section 7 from equivariant u-self maps and the Dugger-Isaksen isomorphism, not assumed.

assumptions (8)
  • standard math Lurie's infinity-category foundations and presentable infinity-categories
    Used throughout Sections 2-5 for limits, colimits, adjunctions, and monoidal Barr-Beck arguments; treated as established background.
  • standard math Monoidal Barr-Beck theorem of Mathew-Naumann-Noel
    Theorem 3.6 is the key device for identifying module categories; cited and applied as a proven theorem.
  • domain assumption Segal conjecture for C2, equivalently Lin's theorem
    Used in Proposition 7.1 to identify C2-equivariant homotopy of spheres with non-equivariant homotopy of stunted projective spectra.
  • domain assumption Bachmann's theorem that real Betti realization is rho-localization, SH(R)[rho^{-1}] equivalent to Sp
    Theorem 8.10 gives the geometric part of the isotropy separation square and the isomorphism on pi^R[rho^{-1}]; external theorem.
  • domain assumption Dugger-Isaksen isomorphism theorem pi^R_{i,j} S is isomorphic to pi^{C2}_{i,j} S for i >= 3j-5
    Theorem 7.6 is load-bearing for lifting C2-equivariant u-self maps to R-motivic tau-self maps at the prime 2.
  • domain assumption Stahn's odd-primary motivic Adams-Novikov computations, including pi^R_{*,*} BPGL and the spectral sequence recipe
    Provides Z_p[tau^2] for the rho-complete sphere and tau^2-self maps at odd primes in Proposition 7.11.
  • domain assumption Landweber's James periodicity theorem and stable stems of stunted projective spectra
    Used to construct u-self maps on C(a_i) and hence tau-self maps in Section 7.
  • domain assumption Convergence of the motivic Adams spectral sequence in the cases used
    Used in the proof of Proposition 8.3 to verify that C(rho) maps to Spec C_+ is a p-complete cellular equivalence; convergence cited from Dugger-Isaksen, Stahn, and Hu-Kriz-Ormsby.

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Pith. "Pith review of C_2-equivariant stable homotopy from real motivic stable homotopy." pith.science (2026). https://pith.science/paper/E5NPRTVD

@misc{pith2026190808378,
  author       = {Pith},
  title        = {Pith review of: C_2-equivariant stable homotopy from real motivic stable homotopy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E5NPRTVD}},
  note         = {Machine review of arXiv:1908.08378}
}
read the original abstract

We give a method for computing the C_2-equivariant homotopy groups of the Betti realization of a p-complete cellular motivic spectrum over R in terms of its motivic homotopy groups. More generally, we show that Betti realization presents the C_2-equivariant p-complete stable homotopy category as a localization of the p-complete cellular real motivic stable homotopy category.

Figures

Figures reproduced from arXiv: 1908.08378 by the authors.

Figure 7.1
Figure 7.1. The structure of π C2 i,j S 0,0 . Theorem 7.6 (Dugger-Isaksen). C2-Betti realization induces an isomorphism π R i,jS 0,0 π C2 i,j S 0,0 for i ≥ 3j − 5 [PITH_FULL_IMAGE:figures/full_fig_p028_7_1.png] view at source ↗
Figure 9.1
Figure 9.1. Computing π C2 ∗,∗HF2 from π R ∗,∗ (HF2)R. 43 [PITH_FULL_IMAGE:figures/full_fig_p043_9_1.png] view at source ↗
Figure 9.2
Figure 9.2. Computing π C2 ∗,∗HZ2 from π R ∗,∗ (HZ2)R. 46 [PITH_FULL_IMAGE:figures/full_fig_p046_9_2.png] view at source ↗
Figures from the paper (1 more)
Figure 9.3
Figure 9.3. Figure 9.3: Computing π C2 ∗,∗kR∧ 2 from π R ∗,∗ (kgl∧ 2 )R. The calculation is displayed in [PITH_FULL_IMAGE:figures/full_fig_p048_9_3.png]

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