{"id":"7e9ef107-f0db-41b3-959e-701425190994","arxiv_id":"2601.11263","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a simply connected space X, coHochschild homology of R[X] with Thom-spectrum coefficient admits a cellular filtration and reduces to a simplicial object built from coHochschild homology of loop groups.","lead":"This paper finds a way to compute a loop-style invariant, called topological coHochschild homology, when the coefficient object is a Thom spectrum built from a simply connected space. It works by chopping the space into cells and by reducing the computation to loop groups.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main results depend on the author's unpublished cotensor identity [24, Theorem 1.1]; Theorem 1.3's proof inherits this load-bearing dependency.","rationale":"The reader's CONDITIONAL verdict is appropriate. The paper is well-structured and the arguments are plausible, but the reliance on [24, Theorem 1.1] is a genuine missing support; the theorem is used in the proofs of Proposition 3.4, Lemma 4.3, and Proposition 4.9, and the filtration theorem's proof explicitly invokes 'a similar argument to Proposition 3.4.' I differ from the reader by prioritizing the cotensor identity as the load-bearing concern rather than the commutation with infinite direct sums: the connectivity argument suppresses the infinite-cell issue, whereas failure of the cotensor identity would collapse the entire computation. Since the reader already made the verdict conditional on addressing such gaps, no change in verdict is warranted.","tokens_in":19296,"tokens_out":16916,"duration_ms":172977,"concrete_test":"Independently derive the cotensor identity coTHH_R(M;C) ≃ M □_{C⊗C^op} C for the case C = R[X] with X simply connected and M = Thf directly from the definition of coTHH as a totalization, and use it to prove the associated graded formula in Theorem 1.3 for an infinite CW complex. If the identity cannot be established without additional hypotheses, the main theorem lacks a complete, self-contained proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing step is the wholesale import of the cotensor product identity coTHH_R(M;C) ≃ M □_{C⊗C^op} C from the author's unpublished preprint [24, Theorem 1.1]. This identity is invoked in the proof of Proposition 3.4, which drives the connectivity argument underlying Theorem 1.2, Theorem 1.3 (via 'a similar argument'), Lemma 4.3, and Proposition 4.9. The paper gives no proof of this identity, and the correctness of the filtration's associated graded computation depends on it. The secondary concern about commuting totalization with infinite direct sums in Theorem 3.8 is likely addressable by the same connectivity argument (the fibers F_n are (n−1)-connective uniformly, so the Milnor-sequence argument goes through for infinite sums), but only if the cotensor identity holds. Thus the unpublished dependency is the single point whose failure would invalidate the central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies topological coHochschild homology (coTHH) of Thom spectra. For an E∞-ring R and a map f : X → BGL1(R) with X simply connected, it claims an exhaustive filtration on coTHH^R(Thf; R[X]) indexed by the cellular skeleta of X, with associated graded ⊕_{X(n)} Σ^n R[ΩX] (Theorem 1.3). It also claims a base-change formula (Theorem 1.2) and a reduction of coTHH^R(Thf; R[X]) to the geometric realization of coTHH of group-ring coalgebras R[G^n] (Theorem 1.4). The proofs use ∞-category machinery and compare the comodule structure with Beardsley's construction.","tokens_in":19534,"tokens_out":3634,"duration_ms":40360,"significance":"If the main theorems are correct, they provide a genuinely new and potentially powerful computational tool: coTHH of a Thom spectrum is computed cell-by-cell from the loop space of X, and it is reduced to coTHH of group rings. The manuscript is carefully written, includes detailed proofs of several structural results (e.g., Proposition 2.10 and Theorem 2.11), and explicitly compares with existing work of Beardsley. However, the central results depend on an identity coTHH_R(M;C) ≃ M □_{C⊗C^op} C imported from the author's own unpublished preprint [24]; this dependency is load-bearing and is not independently verified in the present manuscript. The significance is therefore conditional on the correctness and availability of that companion result.","major_comments":[{"comment":"The proof of Proposition 3.4 begins with 'By [24, Theorem 1.1]', importing the equivalence coTHH_R(M;C) ≃ M □_{C⊗C^op} C. This identity is then used in the proof of Theorem 3.8 (including the 'similar argument' for the direct sum over cells), Lemma 4.3, and Proposition 4.9. Since [24] is an unpublished same-author preprint, this is a load-bearing external dependency. The manuscript should either include a proof of the cotensor equivalence, or state it as a numbered theorem with precise hypotheses and a complete proof, so that the present paper is self-contained on this point. The editor should also confirm the status of [24].","section":"§3, Proposition 3.4"},{"comment":"In the proof of Theorem 3.8, the step 'A similar argument to the proof of Proposition 3.4 shows that coTHH_R(⊕_{X(n)} R; R[X]) ≃ ⊕_{X(n)} coTHH_R(R; R[X])' is not justified. coTHH_R(−;R[X]) is a totalization, and totalization does not generally commute with infinite direct sums. The number of n-cells X(n) may be infinite, so the argument requires a Milnor-sequence/convergence argument showing that the relevant fibers are uniformly (n−1)-connective. The paper does not provide such an argument; it only gestures at Proposition 3.4. This gap is central to the associated graded computation, and the missing argument itself depends on the cotensor identity from [24].","section":"§3, Theorem 3.8"},{"comment":"The identification of each summand in the associated graded uses the equivalence coTHH_R(R; R[X]) ≃ R[ΩX]. The text says this can be deduced from [24, Lemma 3.2] or Proposition 3.4, but no proof is given. Since this equivalence is a key input to the main filtration theorem, a direct proof should be included in the paper, or an explicit proof in the literature should be cited with enough detail for the reader to verify the result under the stated hypotheses (connective R, X simply connected).","section":"§3, Example 3.3 / Theorem 1.3"}],"minor_comments":[{"comment":"Typos: 'parper' in the abstract; 'sence' in the introduction; 'the number of n-cells' should be 'the number of n-cells' without the spurious 'the the'; 'colimt' in the proof of Theorem 3.8.","section":"Abstract and Introduction"},{"comment":"In the proof of Proposition 2.6, 'university property' should be 'universal property'.","section":"§2, Proposition 2.6"},{"comment":"The phrase 'Although N(Δ^{op}_{≤n}) is an infinite simplicial set' is inaccurate: the category Δ_{≤n} is finite, so its nerve is a finite simplicial set. If a different simplicial set was intended, please clarify.","section":"§4, Theorem 4.4"},{"comment":"In the proof, 'Now the left hand of the comparison map f' is confusing; the intended statement is presumably that the domain of f is equivalent to the geometric realization of {coTHH_R(R[G^n])}_n. Please rephrase.","section":"§4, Theorem 4.4"},{"comment":"Several citation typos occur: 'Therorem 2.11' in the proof of Theorem 4.8, and 'simplicical' in §2. Please proofread the references and technical vocabulary.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The central risk is the paper's reliance on the author's own unpublished preprint [24] for the cotensor identity. If [24] is not yet available or not independently verified, the main theorems rest on unverified scaffolding. I recommend that the editor request either a proof of the cotensor identity within this paper or confirmation of the external reference's status before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Eva, quick read of Zha's arXiv:2601.11263. The paper delivers what it promises: a cellular filtration on coTHH^R(Thf; R[X]) for simply connected X (Thm 1.3), and a simplicial reduction to coTHH of group rings (Thm 1.4). These are new as far as I know. Thm 1.2 is the coTHH analogue of Blumberg's THH equivalence, less surprising but still a useful statement. The exposition is solid: the infinity-category machinery is standard, the comodule structure is carefully compared with Beardsley, and the proof strategy is coherent. The paper is a real contribution to the coTHH program.\n\nThe soft spot is not hidden: the load-bearing equivalence is the cotensor identity coTHH_R(M;C) ≃ M □_{C⊗C^op} C, imported from the author's own unpublished preprint [24, Thm 1.1]. That identity drives Prop 3.4, which in turn drives Thm 1.2, Thm 1.3, and Prop 4.9. The present paper does not prove it. If that identity is wrong, the central claims go down with it. This is a genuine dependency, not manufactured. The author should either give a self-contained proof or at least make the dependency explicit and verified; a referee has to check [24] carefully.\n\nA second, smaller gap: in the proof of Thm 1.3, the step that commutes coTHH with the infinite direct sum over n-cells is dispatched with 'a similar argument to the proof of Proposition 3.4' without showing the convergence. For infinite CW complexes this is not automatic. I suspect it can be fixed via the same connectivity/Milnor-sequence argument, but it needs to be written.\n\nMinor: a few typos ('parper', 'proof' for 'prove') and the paper would benefit from stating the finiteness assumptions on the CW structure. None of that affects the mathematics.\n\nBottom line: this is a paper I would send to a serious referee. The novelty is genuine, the proofs are mostly detailed, and the dependency on [24] is identifiable and checkable rather than hidden. My recommendation is conditional accept after the author either proves or carefully states the cotensor identity and fills the totalization-direct-sum gap.\n\nFor a reading group: maybe, if you're in stable homotopy. I'd cite it if I worked on coTHH.","headline":"New filtration and loop-group reduction for coTHH with Thom spectrum coefficients — genuinely useful, but the main results inherit an unproved cotensor identity from the author's earlier preprint.","tokens_in":19992,"tokens_out":3054,"would_cite":true,"duration_ms":31307,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55P43","55P35"],"pacs":[],"model":"deepseek-v4-flash","headline":"Topological coHochschild homology of Thom spectra is computable from the cellular structure of the base space.","keywords":["topological coHochschild homology","Thom spectra","E-infinity ring spectra","comodules","cellular filtration","free loop spaces","cotensor products","spectral sequences"],"falsifier":"Compute coTHH^R(Thf;R[X]) for a space X with infinitely many cells in a fixed dimension—for instance CP∞, which the paper itself treats in Example 3.10, or an infinite wedge of n-spheres—and check whether the filtration's graded pieces match the predicted ⊕ Σ^n R[ΩX]. A mismatch would show that the totalization does not commute with the direct-sum decomposition in the needed generality. A second check is to verify the companion cotensor identity for a small non-finite coalgebra.","tokens_in":19214,"feed_emoji":"🔄","tokens_out":10835,"duration_ms":99404,"temperature":0.7,"pith_summary":"This paper tries to establish that topological coHochschild homology (coTHH) of a Thom spectrum, with coefficients in the corresponding trivial Thom spectrum, can be computed from the cellular structure of the underlying space. For a simply connected space X and a connective commutative ring spectrum R, the R-module coTHH^R(Thf;R[X]) is shown to admit an exhaustive filtration indexed by the skeleta of X, whose n-th graded piece is the direct sum, over the n-cells of X, of Σ^n R[ΩX]. If this is right, a complicated totalization becomes a cell-by-cell computation in terms of the loop space ΩX, and coTHH with Thom coefficients reduces to coTHH of group rings R[ΩX^n]. These results matter because coTHH is the coalgebra-side counterpart of topological Hochschild homology and is tied to free loop spaces, while Thom spectra are central in twisted cohomology.","feed_headline":"Cell-by-cell formula computes coHochschild homology of Thom spectra","feed_subtitle":"New filtration reduces coHochschild homology of a Thom spectrum to pieces built from its loop space.","key_machinery":"The carrying object is the Thom spectrum functor on the slice category over BGL1(R), the space of invertible R-modules. The paper shows this functor is symmetric monoidal, so the coalgebra structure of X induces an R[X]-comodule structure on Thf. Three identities do the work: the cotensor/cobar identity coTHH_R(M;C) ≃ M □_{C⊗C^op} C, imported from the author's companion preprint, which lets one compute coTHH as a cotensor product; the equivalence Thf ≃ R⊗_{R[ΩX]} R, expressing a Thom spectrum as a relative tensor product; and the skeleton filtration of X, which yields the cell-by-cell associated graded.","core_discovery":"The paper's central claim is that, for a connective E-infinity ring R (a commutative ring spectrum) and a simply connected space X with a map f:X→BGL1(R) (the space of invertible R-modules), the R-module coTHH^R(Thf;R[X])—topological coHochschild homology of the Thom spectrum of f with coefficients in the trivial Thom spectrum R[X]=R⊗Σ∞+X—admits an exhaustive filtration indexed by the skeleta of X. The n-th graded piece is a direct sum, one copy for each n-cell of X, of Σ^n R[ΩX]. The computation reduces to a colimit of coTHH of group rings R[ΩX^n] with explicit face and degeneracy maps. This is the coHochschild counterpart of known THH results for Thom spectra: a totalization becomes cell-b","pith_inferences":["Beyond the paper's claims, the filtration in Theorem 1.3 should yield a cellular spectral sequence for coTHH of Thom spectra in a range of cases, making the computation a cell-counting exercise whenever X is finite; this is a direct extension the author does not spell out.","The formula of Proposition 4.9 suggests coTHH with Thom coefficients behaves like a 'twisted free loop spectrum' Thf[ΩX] in general; testing it on examples such as products of spheres or lens spaces would show whether the E1 hypothesis can be weakened.","A caveat located by the paper's own text: the cell-structure step in Theorem 1.3 is justified by 'a similar argument to the proof of Proposition 3.4' rather than a detailed proof, and several key statements depend on the cotensor identity from the author's companion preprint. The main theorems should therefore be read as conditional on that identity and on the totalization/direct-sum commutation."],"forward_implications":["If Theorem 1.3 holds, coTHH of a Thom spectrum can be assembled from the cells of X, with each n-cell contributing a copy of Σ^n R[ΩX]; for finite CW complexes this gives a finite, explicit cell-by-cell computation.","The orientation result (Theorem 1.2) says that when f is E∞-oriented by a commutative R-algebra A, base change turns coTHH^R(Thf;R[X]) into A⊗coTHH(Σ∞+X), linking the invariant to ordinary free loop spectra.","Theorem 1.4 reduces coTHH^R(Thf;R[X]) to a colimit of coTHH of group rings R[ΩX^n], with face maps induced by the G-action, multiplication, and augmentation of the loop group.","Corollary 3.9 gives a spectral sequence whose E^1 page is a direct sum over p-cells of π_{p+q}(Σ^p A[ΩX]), converging to the A-homology of coTHH^R(Thf;R[X]).","For a simply connected E1-space (a homotopy associative monoid), Proposition 4.9 gives the compact formula coTHH^R(Thf;R[X]) ≃ Thf[ΩX], a free-loop-style description."],"fun_headline_variants":["Cell filtration computes coHochschild homology of Thom spectra","Thom spectra: cell-by-cell coTHH formula","coTHH of Thom spectra reduced to loop spaces","New filtration for coHochschild homology of Thom spectra","Cell-by-cell decomposition for coTHH of Thom spectra"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing assumption is that the totalization defining topological coHochschild homology can be interchanged with the direct sums over the cells of X, and that the cotensor identity taken from the author's companion preprint holds for the relevant coalgebras; if either fails, the filtration theorem may only hold under additional finiteness hypotheses.","fun_headline_variants_meta":{"raw":{"variants":["Cell filtration computes coHochschild homology of Thom spectra","Thom spectra: cell-by-cell coTHH formula","coTHH of Thom spectra reduced to loop spaces","New filtration for coHochschild homology of Thom spectra","Cell-by-cell decomposition for coTHH of Thom spectra"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000151,"raw_usage":{"total_tokens":1040,"prompt_tokens":748,"completion_tokens":292,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":492,"completion_tokens_details":{"reasoning_tokens":213}},"tokens_in":492,"tokens_out":292,"duration_ms":3327,"temperature":1.0,"reasoning_tokens":213,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T10:04:59.502397+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute coTHH^R(Thf;R[X]) for a space X with infinitely many cells in a fixed dimension—for instance CP∞, which the paper itself treats in Example 3.10, or an infinite wedge of n-spheres—and check whether the filtration's graded pieces match the predicted ⊕ Σ^n R[ΩX]. A mismatch would show that the totalization does not commute with the direct-sum decomposition in the needed generality. A second check is to verify the companion cotensor identity for a small non-finite coalgebra.","supporting_citations":[],"review_version":1}