{"id":"d81a3f17-34fa-44ff-9d90-e4806ab26080","arxiv_id":"1908.11321","paper_version":4,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new Hecke spectral sequence computes the Morava E-theory of unordered configuration spaces, yielding explicit E-theory and F_p-homology results for p-point configurations.","lead":"This paper constructs a new spectral sequence that computes the Morava E-theory of configuration spaces, and uses it to calculate these invariants for configurations of p points in Euclidean spaces and on punctured surfaces. It matters because these computations determine operations on iterated loop spaces and test a conjecture of Ravenel about Morava K-theory.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"E2-page identification rests on an unproved completion-to-isomorphism upgrade in Cor. 4.21; the main theorems are conditional on it.","rationale":"The reader located the same general area: Brantner's theorem. I agree it is the weakest assumption, but I want to be more precise: the paper needs Corollary 4.21, not merely the completion statement, and it needs that upgrade at every level of a monadic bar construction. That precision is why I mark agreement as partial. The computations in Sections 6 and 7 are detailed and the height-1 checks are genuine independent evidence, so this is not a rejection. However, if Corollary 4.21 is not independently verifiable in the stated positive-weight finite-free case, then the central E2-page identification has a gap. A CONDITIONAL acceptance—contingent on supplying the proof of Corollary 4.21 or a published reference—is the appropriate verdict.","tokens_in":56874,"tokens_out":17353,"duration_ms":177831,"concrete_test":"Verify Corollary 4.21 in the minimal case actually used in Theorem 5.5: take M = E{1} (one free generator in weight 1), compute pi_*FreeLE(LK(h)M) from the explicit formulas of [Bra17, §4.4.2] and the free Hecke Lie algebra L_Hu(E_*{1}), and check they are isomorphic before completion, including the Hecke operations. Then recompute the Euclidean case (n,k)=(1,2) of Theorem 6.10 from the E2-page of Theorem 5.5: the Hecke homology must be a single free E_*-class and the spectral sequence must collapse, matching the James-splitting description of the weight-p summand of Omega S^3. Agreement would settle the concern; a mismatch would show the upgrade or its bar-level application is unsound.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"I found no internal inconsistency in the paper's own algebra (Sections 2–4). The load-bearing soft spot is the unproved upgrade from Brantner's Theorem 4.20 to Corollary 4.21. Theorem 4.20(2) identifies FreeLie_Hu(pi_*M) with pi_*FreeLE(LK(h)M) only after completion; Corollary 4.21 asserts an honest isomorphism for positive-weight finite-free E-modules, and the proof of Theorem 5.5 applies this assertion levelwise to replace the homotopy of the topological bar construction by the algebraic bar Bar(id,L_Hu,g(M;X)) and then by CE_Hu(g). This step is essential: it is exactly what turns the bar spectral sequence of Theorem 5.1 into a spectral sequence whose E2-page is Hecke Lie algebra homology. The paper does not reproduce the argument for the upgrade; the one-sentence reference to [Bra17, §4.4.2] is not a proof of target-side finiteness and naturality of the cotensor construction. The height-1 comparison in §6.1 gives independent support but only checks cases in which the algebraic and completed answers coincide, so it does not settle the general upgrade.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops a new method for computing the completed Morava E-homology of unordered configuration spaces of framed manifolds. The main structural result, Theorem 5.5, is a weighted spectral sequence converging to ⊕_k E^∧_{s+t}(B_k(M;X)), whose E2-page is identified with the homology of a Hecke–Chevalley–Eilenberg complex for an unshifted Hecke Lie algebra g(M;X). The construction combines Knudsen's stable splitting of configuration spaces (Theorem 5.1) with Brantner's description of the E-homology of free K(h)-local spectral Lie algebras in terms of Hecke Lie algebras (Theorem 4.20). Using this tool, the paper computes the E-theory of the weight-p Snaith summands of iterated loop spaces of spheres at arbitrary height (Theorem 6.10), derives the corresponding Morava K-theory groups (Theorem 6.19), and computes the E-theory and F_p-homology of configuration spaces of p points on punctured surfaces (Theorems 7.1 and 1.10). The paper also proves several foundational algebraic results, including the Chevalley–Eilenberg computation for flat Lie algebras (Theorem 3.13) and the existence of transferred model structures for Hecke Lie algebras.","tokens_in":57152,"tokens_out":7594,"duration_ms":65661,"significance":"If the identification of the E2-page is fully justified, the paper supplies a general and explicit computational tool for Morava E-theory of labeled configuration spaces, with applications to power operations on E_n-algebras and to longstanding conjectures of Ravenel. The paper is carefully written and contains complete proofs of the algebraic foundations in Sections 2–4, including the Chevalley–Eilenberg theorem for flat Lie algebras and the construction of the Hecke–Chevalley–Eilenberg complex. The derived computations are explicit and falsifiable, and they match independent results at height 1 (Langsetmo) and classical F_p results (Cohen). The main gap is the unproved upgrade from a completion isomorphism to an honest isomorphism in Corollary 4.21, which is the load-bearing bridge to the spectral sequence.","major_comments":[{"comment":"The E2-page identification of the Hecke spectral sequence rests on the assertion that the canonical map FreeLie_Hu(π_* M) → π_* FreeLE(L_K(h) M) is an isomorphism for M in the category of pointwise finite free positive-weight E-module spectra, whereas Theorem 4.20(2) only provides this isomorphism after completion. Corollary 4.21 is stated without proof, with only a reference to [Bra17, §4.4.2]. This upgrade is load-bearing: Theorem 5.5 applies it levelwise to replace the homotopy of the topological bar construction by the algebraic bar construction Bar(id, L_Hu, g(M;X)) and then by CE_Hu(g). The height-1 comparison in §6.1 gives indirect evidence but does not test cases in which the completed and uncompleted answers differ. I recommend adding a proof of Corollary 4.21, or at least a detailed account of the finiteness and naturality of the cotensor construction that yields the honest isomorphism.","section":"Section 4.5, Corollary 4.21; used in Section 5.2, proof of Theorem 5.5"}],"minor_comments":[{"comment":"The expression \"Bar•(LHu, ,LHu, g(M;X))\" contains an extra comma, and the reference to \"Proposition 4.21\" should be \"Corollary 4.21.\"","section":"Section 5.2, proof of Theorem 5.5"},{"comment":"In the discussion of the map E2_0,*(p,1,k) → E2_0,*(p,n,k) for n odd and k even, the map is written as multiplication by p^{n/2}, but n is odd in that case; the exponent should be floor(n/2) or the description of the induced map on the torsion subgroups should be clarified.","section":"Section 6.1, proof of Theorem 6.1"},{"comment":"The formula for E∗(Bp( ˙T ))) has an extra closing parenthesis; it should be E∗(Bp( ˙T )).","section":"Section 7.1, Theorem 7.1(1)"},{"comment":"The expression \"α 5\" should presumably be \"α^5\"; the exponent is missing a caret.","section":"Remark 1.4"}],"recommendation":"major_revision","confidential_remarks":"The paper is a strong contribution and the computations appear credible. The central issue is the reliance on the unproved Corollary 4.21; I recommend asking the authors to include a proof or a precise reference that establishes the honest isomorphism. I found no evidence of circularity, since the computed invariants are not used in the definitions of the objects that feed into the spectral sequence."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the deal. This paper delivers a genuinely new computational tool: a spectral sequence for Morava E-homology of labeled configuration spaces, with E2 identified as Hecke Lie algebra homology, and uses it to compute weight p pieces in Euclidean space and punctured surfaces, including a new no-p-torsion result for B_p(S_{g,1}). If the main theorem holds, this is a significant advance. The paper is also strong on the algebraic side: it gives a complete proof of the flat Chevalley-Eilenberg statement (Thm 3.13), explicit complexes, honest discussion of overlaps with Langsetmo and Cohen, and concrete height-2 examples.\n\nThe soft spot is the one the stress-test note flags, and I think it's real. The bridge from the topological bar construction to Hecke Chevalley-Eilenberg homology goes through Cor 4.21, which upgrades Brantner's Theorem 4.20 from an isomorphism after completion to an honest isomorphism for finite-free positive-weight modules. That upgrade is asserted with a one-line reference to the thesis. The proof of Theorem 5.5 applies it levelwise; without it, you'd get a completed E2 page, not the stated E2 page. So the main theorem is conditional on an unproved technical statement. I don't expect the statement to be false—the height-1 checks and the general shape make it plausible—but it is exactly the kind of thing referees need to see proved or precisely cited to a verifiable source.\n\nOther concerns are minor: reliance on Knudsen's stable splitting is fine (published and apparently correct), and there are a few typos (Thm 7.1 has a doubled parenthesis; Thm 5.5 cites 'Proposition 4.21' instead of Cor 4.21). Not circular: the computed invariants don't feed back into the definitions.\n\nWho's this for? Chromatic homotopy types and anyone computing with power operations; also topologists interested in F_p homology of surfaces. It deserves peer review at a serious journal, with a request to fix the gap in Cor 4.21 or make the theorem truly independent. Overall: major revision or conditional accept, but not a desk reject.","headline":"Genuinely new computational machinery for Morava E-homology of configuration spaces, but the main theorem rests on an unproved completion-to-isomorphism upgrade from the first author's thesis.","tokens_in":57660,"tokens_out":2441,"would_cite":true,"duration_ms":23883,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55R80","55P48","55N20","55T99"],"pacs":[],"model":"deepseek-v4-flash","headline":"A convergent spectral sequence computes Morava E-theory of labeled configuration spaces, with its E2 page identified as the homology of a Hecke–Chevalley–Eilenberg complex.","keywords":["Morava E-theory","configuration spaces","Hecke Lie algebras","spectral sequences","Lubin-Tate theory","iterated loop spaces","Dyer-Lashof operations","surface configurations"],"falsifier":"Compute the completed E-homology of $B_3(\\mathbb{R}^3)$ at height 2 and p=3 by an independent method, such as a Gysin sequence for the symmetric group action or a direct Atiyah–Hirzebruch computation, and compare with the paper's prediction $E_* \\oplus \\Sigma^{-1}E_*/3$; a different torsion summand would falsify the spectral sequence's E2-page identification.","tokens_in":56697,"feed_emoji":"🧮","tokens_out":7764,"duration_ms":69253,"temperature":0.7,"pith_summary":"This paper introduces a spectral sequence that computes the Morava E-theory of unordered configuration spaces at odd primes and any chromatic height. The E2 page is identified with the homology of a Chevalley–Eilenberg-style complex built from Hecke Lie algebras, algebraic objects that package the additive operations acting on completed E-homology. From this general tool the authors derive explicit formulas for the E-theory of the weight p summands of iterated loop spaces of spheres and of configurations of p points on punctured surfaces. These computations yield Morava K-theory groups relevant to a longstanding conjecture on loop spaces and, in the limit as height tends to infinity, new results on ordinary mod p homology of configuration spaces of surfaces. The paper's central claim is that this spectral sequence exists, has the stated algebraic E2 page, and is powerful enough to make these computations routine.","feed_headline":"Spectral sequence computes E-theory of configuration spaces","feed_subtitle":"Its E2 page is Hecke Lie algebra homology, yielding explicit p-point and loop-space formulas at all heights.","key_machinery":"The load-bearing object is the Hecke Lie algebra $g(M;X) = E^{\\wedge}_*(\\mathrm{FreeLie}(\\Sigma^{n-1}X)^{M_+})$, together with its Hecke–Chevalley–Eilenberg complex $CE_{Hu}$. A Hecke Lie algebra is a Lie algebra in $E_*$-modules equipped with an action of the ring of additive power operations, and $CE_{Hu}$ is the complex obtained by killing those operations in a derived fashion and then forming the derived abelianisation. The argument converts a stable equivalence between configuration spaces and the bar construction on a free spectral Lie algebra into an algebraic spectral sequence, using a theorem from the first author's thesis that identifies the completed E-homology of a free K(h)-local spectral Lie algebra with the free Hecke Lie algebra on its homotopy groups.","core_discovery":"The paper's central claim is that for a framed n-manifold M and a spectrum X, whenever the completed E-homology Hecke Lie algebra $g(M;X)$ is finite and free in each weight, there is a convergent weighted spectral sequence $E^2_{s,t} \\cong H_{s+1}(CE_{Hu}(g(M;X)))_{t-1}$ converging to $\\bigoplus_{k\\ge 0} E^{\\wedge}_{s+t}(B_k(M;X))$, where $B_k(M;X)$ is the labeled unordered configuration space. Here $CE_{Hu}$ is a Hecke analogue of the Chevalley–Eilenberg complex: it first kills the additive operations in a derived fashion and then takes the derived abelianisation of the resulting Lie algebra. The authors apply this spectral sequence to compute the completed E-homology of the pth Snaith summand of $\\Omega^n S^{n+k}$ for all n, k, and all heights, and the E-cohomology of the space of p unordered points in a punctured orientable surface. They read off the corresponding Morava K-theory groups and, by letting the height grow, prove that the integral homology of $B_p(S_{g,1})$ has no p-power torsion.","pith_inferences":["The same spectral sequence, with a more careful treatment of the coherent coalgebra structure, should extend to weights $p^i$ and could settle the full conjecture at all weights.","The torsion summand of the form $E_*(B\\Sigma_p)/(tr)$ suggests that weight p E-theory of configuration spaces is governed by the modular representation theory of the symmetric group, a connection the paper leaves mostly implicit.","A direct testable extension is to apply the method to non-framed manifolds after incorporating tangential structure groups, following the authors' own stated future direction, and to compare the results against known rational Betti numbers.","For explicit height 2 examples, the formulas give concrete answers such as the p=3 computation $E^{\\wedge}_*(B_3(\\mathbb{R}^{11})) \\cong E_* \\oplus \\Sigma^{-1}E_*[\\alpha]/(\\alpha^4 - 6\\alpha^2 + (h-9)\\alpha - 3, \\alpha^5)$, which could be checked independently through a direct analysis of Dyer–Lashof operations."],"forward_implications":["The E-theory of the pth weight summand of $\\Omega^n S^{n+k}$ is now known explicitly for every n, k, and height h at odd primes, giving a concrete description of the weight p operations on $E_n$-algebras.","For punctured orientable surfaces, the E-cohomology of $B_p(S_{g,1})$ is a direct sum of free $E_*$-modules with explicit ranks, so the E-theory of these configuration spaces is torsion-free.","The same surface computation implies that the integral homology of $B_p(S_{g,1})$ has no p-power torsion, so its $\\mathbb{F}_p$ Betti numbers coincide with its rational Betti numbers.","At height 1 the spectral sequence recovers known p-adic K-theory computations, and as h tends to infinity it yields new $\\mathbb{F}_p$-homology computations for configurations of p points on surfaces.","The Morava K-theory groups read off from the spectral sequence show that the size of the weight p part of $K(h)_*(\\Omega^n S^{n+1})$ depends on height for n at least 4, which runs against the pattern suggested by earlier conjectures."],"supporting_citations":[{"why":"Supplies the theorem that identifies the completed E-homology of free K(h)-local spectral Lie algebras with free Hecke Lie algebras, the bridge underlying the E2-page identification.","marker":"[Bra17]"},{"why":"Establishes the stable equivalence between configuration spaces and the bar construction on a free spectral Lie algebra used as the topological input.","marker":"[Knu18]"},{"why":"Provides the rational Betti number computation used to identify the E2-page in surface cases and to prove degeneration after inverting p.","marker":"[Knu17]"},{"why":"Computes Betti numbers of configuration spaces of surfaces, giving the explicit rank formulas matched by the E-theory computation.","marker":"[DCK17]"},{"why":"Develops the rings of additive power operations whose action defines Hecke modules.","marker":"[Rez09]"},{"why":"Provides finiteness and freeness of completed E-homology of symmetric groups, underwriting the finite-free hypothesis on the Hecke Lie algebra.","marker":"[Str98]"},{"why":"States the conjecture about Morava K-theory of iterated loop spaces that the weight p computations address.","marker":"[Rav98]"},{"why":"Computes rational cohomology of configuration spaces of surfaces, used as input for the surface E-theory computation.","marker":"[BC88]"},{"why":"Gives explicit height 2 power-operation formulas used for the concrete p=3 examples.","marker":"[Zhu14]"}],"fun_headline_variants":["Hecke Lie algebra homology drives config space E-theory","E-theory of unordered configs from Hecke complex","Weighted spectral sequence for loop-space E-theory","No p-torsion in integral homology of p-point configs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole computation rests on an unproved theorem quoted from the first author's thesis: the completed E-homology of a free spectral Lie algebra is exactly the free Hecke Lie algebra on its homotopy groups, and the paper does not reprove it.","fun_headline_variants_meta":{"raw":{"variants":["Hecke Lie algebra homology drives config space E-theory","E-theory of unordered configs from Hecke complex","Weighted spectral sequence for loop-space E-theory","No p-torsion in integral homology of p-point configs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000569,"raw_usage":{"total_tokens":2695,"prompt_tokens":947,"completion_tokens":1748,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":563,"completion_tokens_details":{"reasoning_tokens":1680}},"tokens_in":563,"tokens_out":1748,"duration_ms":12035,"temperature":1.0,"reasoning_tokens":1680,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:17:54.783924+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the completed E-homology of $B_3(\\mathbb{R}^3)$ at height 2 and p=3 by an independent method, such as a Gysin sequence for the symmetric group action or a direct Atiyah–Hirzebruch computation, and compare with the paper's prediction $E_* \\oplus \\Sigma^{-1}E_*/3$; a different torsion summand would falsify the spectral sequence's E2-page identification.","supporting_citations":[],"review_version":1}