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REVIEW 3 major objections 4 minor 27 references

Level structures and cyclic power operations on the homology of $\mathbb{E}_\infty$ spaces

T0 review · 3 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read This paper derives explicit finite-sum formulas for the additive C2 power operation on the E-homology of BU and BSU for complex K-theory and height-2 Morava E-theory, and shows that at height 2 the indecomposables of E2[BSU] form a module t

desk verdict Useful new computations in power operations, but the height-2 formulas rest on one unproved identity from Schumann's thesis; refereeing should require verification. read the letter →

arxiv 2607.18670 v1 pith:ZNSRFT4R submitted 2026-07-21 math.AT

Catherine Li This is my paper · ORCID
classification math.AT MSC 14L0555N2055N2255P4355S12
keywords poweroperationsMoravaE-theorylevelstructuresformalgroupsBUhomologyBSUchromatichomotopytheorycyclic
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 works out explicit formulas for the additive C2 power operation on the homology of the infinite loop spaces BU, BU×Z, and BSU, for two complex-oriented theories: complex K-theory (height 1) and a height-2 Morava E-theory at p=2. The driving idea is to read power operations through level structures on formal groups: the power operation pulls back a universal map to the multiplicative group, and the answer becomes a product over the level structure. For K-theory the paper recovers the known operation θ, and for height 2 it derives closed formulas for the three component operations Q0, Q1, Q2 on the indecomposables of E2[BU]. Applying these to BSU, the paper finds that after killing the ideal (θ,Q1,Q2,2,a) the module of indecomposables is F2{d2}⊕F2{d3}, a structural contrast with the height-1 case and evidence that the module is finitely generated.

What carries the argument

The key objects are level C2 structures on formal groups: a chosen point of order 2 whose coordinate z satisfies the 2-series relation. On an E∞ ring E, the additive C2 power operation factors through E0(BC2)/I_tr, which is isomorphic to the scheme of level-2 structures on the formal group of E. The computational engine is Theorem 3.3: the power operation on the universal pointed map h(x)=1+b1x+b2x2+… to G_m is the product ∏_{a∈C2} h(x+_G x(ℓ(a))) / h(x(ℓ(a))). Expanding this identity and matching coefficients, using a standard combinatorial identity for power sums and, at height 2, a computationally verified identity for the coordinate of the sum of a point with the universal 2-torsion poin

What would settle it

Evaluate the identity w=(d-u)/(1+d^2 u) numerically or symbolically for a few points P and Q on the universal deformation of the supersingular curve, using the elliptic curve group law, and check that the result satisfies the curve's equation and the 2-torsion condition; a single counterexample would invalidate the central formulas and the finite-generation conclusion.

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Extended reading notes

Core claim

The central claim is that the additive C2 power operation on E2[BU] decomposes uniquely as P(x)=Q0(x)+Q1(x)d+Q2(x)d2, and the paper proves explicit finite-sum formulas for θ(bn), Q1(bn), and Q2(bn) modulo decomposables, in terms of the generators bi and the parameters a and 2 (Theorem 5.2, equations (1)–(3)). The same computation yields formulas for E2[BSU] (Corollary 5.1). A striking consequence is that for the module of indecomposables M of E2[BSU], the quotient F2 ⊗Δ M is F2{d2}⊕F2{d3}; the paper interprets this as evidence that M is finitely generated as a Δ-module, and speculates that E2[BSU] is K(2)-locally finite-celled as an E∞ E2-algebra, in contrast to the height-1 case.

Load-bearing premise

The height-2 computation rests on a single computationally verified identity for the coordinate of the sum of a point with the universal order-2 point on the supersingular curve; if that identity is wrong or fails under hidden hypotheses about the formal group, the formulas (1)–(3) and their consequences do not follow.

Editorial extensions

If this is right

  • For complex K-theory, the formula θ(bn)=b2n+b2n+1 mod 2 (and the full integral formula) determines the additive operation on indecomposables, agreeing with prior results in the literature.
  • At height 2, the explicit formulas for θ, Q1, and Q2 allow one to compute the Δ-module structure on the indecomposables of E2[BU] and E2[BSU] at p=2, giving a concrete starting point for studying the module of operations.
  • The collapse F2 ⊗Δ M ≅ F2{d2}⊕F2{d3} suggests M is finitely generated as a Δ-module; if true, E2[BSU] would be K(2)-locally finite-celled, contrasting with the non-finite behavior of KU[BSU].
  • The method shows how level structures on formal groups can be used to compute power operations on homology of E∞ spaces without rederiving Quillen's universal calculation from scratch.

Reading between the lines

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

  • The same level-structure method should extend to other heights and primes (e.g., height 2 at p=3 using different elliptic curve models) to produce analogous explicit formulas, potentially revealing a pattern in how the operations depend on the formal group law.
  • The apparent mod-2 splitting of M (where Q1(dn)=Q2(dn)=0 for n a power of 2) may indicate an internal periodicity in the Δ-module structure, possibly reflecting a cellular filtration on BSU.
  • If finite generation of M can be lifted to an explicit resolution, it might yield a new family of K(2)-local finite-celled E∞ algebras, with consequences for the structure of K(2)-local homotopy theory.
  • The formulas depend polynomially on the parameter a, so specializing a to specific values (e.g., a=0) could provide a quick check of the computations and illuminate the transition between different formal group laws.
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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 develops an algebro-geometric framework, based on level structures and the work of Ando–Hopkins–Strickland and Rezk, for computing cyclic power operations on the homology of BU, BU×Z, and related E∞ spaces. The main application is explicit computation, at p=2, of the additive C2 power operation on the indecomposables of KU0(BU) and (E2)0(BU), and subsequent formulas for (E2)0(BSU). The central height-2 result, Theorem 5.2, expresses θ(b_n), Q1(b_n), Q2(b_n) as explicit finite sums, from which a computation of F2⊗Δ M for M = Q((E2)0(BSU)) is derived.

Significance. If correct, this paper would provide one of the few explicit computations of power operations on homology in the Morava E-theory setting, complementing Rezk's and Zhu's cohomological calculations. The derivation is genuinely synthetic: it uses known external inputs (Quillen's formula, Rezk's model, the elliptic curve group law) rather than fitted parameters, and the height-1 formulas are independently checked against Reeker's results. The algebro-geometric interpretation of operations on homology is a useful conceptual contribution. However, the central height-2 computation rests on an unproved and, as stated, apparently inconsistent identity from Schumann's thesis, so the significance of the paper as a whole is currently conditional.

major comments (3)
  1. [§5.2, Fact 5.2] The proof of Theorem 5.2 hinges on the identity w=u(P+Q)=(d-u)/(1+d^2u). This is load-bearing: it is used to derive u+w=d+d^2u', which is essential for the Waring-formula expansion. As stated, however, the identity is inconsistent with the coordinate conventions. In any formal group with coordinate u, the series u(P+Q)=F(u,d) has linear coefficient 1 in u. The displayed rational function has linear coefficient -(1+d^3)=-(ad+3), which is not 1 in Z2[[a]][d]/(d^3-ad-2). For example, at a=0 the coefficient is 3, not 1. Thus the quoted formula cannot be the coordinate of P+Q in the coordinate u=X/Y described in the paper. This indicates a sign or coordinate mismatch in the citation to Schumann. The paper states that no more theoretical proof is known, but the computational identity is not reproduced and no alternative verification is supplied. The author must either prove the identity in the
  2. [§5.2, proof of Theorem 5.2 after Fact 5.2] Even assuming the identity, the passage from u+w=d+d^2u' to the displayed formulas is rapid. The treatment of the boundary case n+i-3j≤0 is correct in substance, but the verification that the only nonzero contribution is the j=n term of [b_{2n}]P(b_n) is compressed. Please expand this step so that a reader can check the indexing without reconstructing the argument. This is secondary to the Fact 5.2 issue, but it would improve confidence in the final formulas.
  3. [Remark 5.1] The claim that F2 ⊗Δ M ≅ F2{d2}⊕F2{d3} is presented with only a sketch: 'it is not too hard to see' and 'from these observations and the commutation relations of [Rez08], it follows.' This is a concrete stated result, not merely a suggestion. Please provide a more detailed derivation, including the relevant commutation relations from [Rez08] and the verification that d2 and d3 do not vanish in the tensor product. The current level of detail is insufficient for a published claim.
minor comments (4)
  1. [Throughout] The notation '2t' in Corollary 5.1 is evidently meant to be '2^t' (powers of two). Several such superscripts appear to be lost in the typesetting, making the formulas hard to read.
  2. [Introduction, Table 1] The table displaying the mod (2,a) reductions is hard to parse; the alignment of the columns and the entries is ambiguous. Please ensure the superscripts and row breaks are clear.
  3. [§2.2] The phrase 'level structures can be understood through algebraic geometry' is intuitive, but the paper would benefit from a precise statement of which of the two quotient rings (by [p](z) vs ⟨p⟩(z)) is used for the target of the additive operation. The distinction is mentioned but could be spelled out.
  4. [§3] The diagrams in the proof of Theorem 3.1 are informative but somewhat informal. For instance, the pullback diagram in rings would be clearer with the maps explicitly labeled by the universal property being used.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper derives power-operation formulas from external inputs (Quillen, Rezk, Schumann) by algebra, without assuming its target formulas or renaming fits as predictions.

full rationale

The derivation chain is not circular. Section 3 reduces the homology power operation to Quillen's coordinate computation P^{CP∞}(x)=∏(x+_G[a]_G(z)), an external input. Theorem 5.1 then uses the multiplicative formal group law and Waring's formula, with an independent check against Reeker's formulas. Theorem 5.2 uses Rezk's model and coordinate computation u'=-u·u(P+Q) together with Schumann's coordinate identity w=(d-u)/(1+d^2u) (Fact 5.2), then derives the explicit formulas for θ, Q1, Q2 purely by coefficient matching, the d^3=ad+2 recurrence, and the generating-function computation for D_m. No free parameter is fitted and no close relative of the target formula is assumed as input. The paper explicitly flags Fact 5.2 as an unproved computational identity ('We are not currently aware of a more theoretical way to prove this surprising identity'), so the height-2 results depend on an external lemma whose proof is not reproduced; this is a rigor/correctness risk, not a circularity, because the lemma is a premise used to compute the formulas, not a conclusion secretly equivalent to them. There is also no load-bearing self-citation: all substantive inputs are external works (Quillen, Rezk, Schumann, Laures, Reeker), and the paper's own conclusions are not used to justify themselves.

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

The paper introduces no invented entities and fits no free parameters. It rests on standard algebraic identities and on published computational inputs from Rezk and Schumann; the latter is the most fragile external dependence.

assumptions (6)
  • standard math Waring's formula (Fact 5.1): A^j+B^j = Σ_k [C(j-k,k)+C(j-k-1,k-1)](-AB)^k(A+B)^{j-2k}.
    Used in proofs of Theorem 5.1 and Theorem 5.2 to expand power sums in terms of elementary symmetric polynomials.
  • domain assumption Level-structure corepresentation: for a 2-periodic even E∞ ring R with p not zero divisor, level(C_p, G_R) ≅ Spf(R^0(BC_p)/I_tr).
    Quoted from HKR00 (Lemma 5.7, Remark 6.15) and used throughout Section 3 to turn power operations into isogenies.
  • domain assumption Rezk's coordinate power operation for height-2 Morava E-theory: u' = P^{CP∞}_E(u) = -u·u(P+Q), where Q is the universal 2-torsion point.
    External input from Rezk (2008), used in the proof of Theorem 5.2 to compute the coordinate part of the power operation.
  • domain assumption Schumann's identity (Fact 5.2): w = u(P+Q) = (d-u)/(1+d^2 u).
    Quoted from Schumann's thesis; the paper states no theoretical proof is known. This identity is load-bearing for Theorem 5.2.
  • domain assumption The Thom isomorphism M P[BU×Z] ≅ M P⊗M P is an E∞ equivalence (Section 3.1.2).
    Used to transfer the power-operation formula from M P⊗M P to M P[BU×Z], and hence to E[BU].
  • domain assumption For KU with multiplicative formal group law x+_G z = x+z-xz, the additive C_2 power operation on the coordinate satisfies P(x)=x(x+_G z).
    Stated in the proof of Theorem 5.1, referencing the E∞ orientation on KU; used to derive the KU[BU] formulas.

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Pith. "Pith review of Level structures and cyclic power operations on the homology of $\mathbb{E}_\infty$ spaces." pith.science (2026). https://pith.science/paper/ZNSRFT4R

@misc{pith2026260718670,
  author       = {Pith},
  title        = {Pith review of: Level structures and cyclic power operations on the homology of $\mathbbE_\infty$ spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZNSRFT4R}},
  note         = {Machine review of arXiv:2607.18670}
}
abstract

In this document, we discuss how cyclic power operations for periodic complex bordism can be understood through algebraic geometry, following from the theory of power operations for Morava $E$-theory developed by Ando, Hopkins, and Strickland. Using this perspective, we describe how one computes power operations on the $E$-homology of $BU$ and $BU\times \mathbb{Z}$, where $E$ is a 2-periodic even $\mathbb{E}_\infty$ ring. Then we provide some explicit example computations at $p=2$; in particular, we compute the action of the additive operations $\Delta$ on the indecomposables $\widehat{Q}(E_0(BU))$ for $E=KU, E_2$.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

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