{"id":"141cc3b0-24aa-4d29-a2a2-5041e22c8bf2","arxiv_id":"2508.20629","paper_version":3,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Under mild axioms on a graded E2-algebra over a positive-characteristic field, higher-order homological stability maps exist with slopes approaching 1, and the patterns are governed by a stability Hopf algebra.","lead":"This paper introduces a chromatic framework for higher-order homological stability, proving that for graded E2-algebras over a field of positive characteristic satisfying three explicit axioms, higher-order stabilisation maps exist with slopes approaching 1. It defines stable homology as a Bousfield localisation and shows that a Hopf algebra constructed from the algebra encodes the possible stability patterns.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The slope quantization k/(k+1) in Theorem A rests on an unproved classification of non-nilpotent generators in Section 5.1; this omitted calculation should be independently verified.","rationale":"The reader identified Axiom (SCE) as the weakest assumption. I do not think that is the most load-bearing concern: (SCE) is an explicitly stated hypothesis, and the theorem is formally conditional on it. The more serious issue is an omitted proof inside the argument for the central conclusion. Section 5.1's classification of the non-nilpotent generators X_{*,*,*} is the step that forces all slopes to be k/(k+1), and it is asserted without proof. This classification feeds directly into Lemma 5.3 and from there into the ordering of the x_i, the construction in Theorem 5.6, and Theorem A's quantitative claims. The rest of the proof — the spectral sequence argument for permanent cycles, the vanishing estimates in Lemma 5.7, and the thick-subcategory arguments in Section 6 — is presented in enough detail that I did not find a separate flaw there. My recommendation is therefore CONDITIONAL rather than a rejection: the main framework and the conditional theorem are credible, but the numerical slope quantization should be checked by an independent computation or a written derivation before the paper is accepted at face value. This does not challenge the novelty or the general architecture, and I agree with the reader's overall positive assessment of the paper's significance.","tokens_in":70942,"tokens_out":14308,"duration_ms":149252,"concrete_test":"Implement, for p = 2 and p = 3 and a small finite-dimensional input W = ⊕_{n≤N} HE2_{n,n−1} (e.g. one-dimensional in degrees 1 through 4), the free W1-algebra basis described in [14, Section 16.2], and list all non-nilpotent multiplicative generators in tridegrees (n,d,f) with d < n. Verify that every such generator has d/n equal to (n−1)/n for p = 2, and equal to (n−1)/n or (n−2)/n with n even for p = 3, and that the lexicographic order on (d/n, f/d) is total on the nonzero components. If a generator with any other slope appears, Theorem A(i) and the quantisation claim must be revised.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem A's quantitative content — that every stabilisation slope is of the form k/(k+1), that the slopes are totally ordered, and hence that vanishing lines quantise — depends on the classification, in Section 5.1, of the non-nilpotent multiplicative generators X_{*,*,*} of π_{*,*,*}(Cτ⊗fil∗R). The text says it is 'an elementary but laborious calculation' that these generators occur only in tridegrees with D = N−1 for p = 2, or D = N−1 or D = N−2 with N even for p odd; this is then used in Lemma 5.3 to identify the possible slopes and to define the total order ≺ used throughout Theorem 5.6 and Theorem A. No proof or precise reference is supplied for this calculation: [14, Section 16.2] describes a basis for the W1-algebra, but the restricted claim about which generators lie below the diagonal and are non-nilpotent is not derived there. This is a load-bearing missing proof. If the omitted calculation is wrong, or if it misses a family of non-nilpotent generators with other slopes, then Theorem A(i), the quantisation of vanishing lines in §1.5.1, and the numerical form of the stability Hopf algebra slogan all fail, even though the rest of the induction in Section 5 could remain intact. This concern is distinct from the reader's SCE worry: (SCE) is an explicit hypothesis, whereas the generator classification is asserted as a fact in the proof of the main theorem.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a chromatic framework for higher-order homological stability. The basic datum is a graded E2-algebra R over a field k, and a higher-order stability theorem is formalised as the existence of an iterated cofibre, or Smith–Toda complex, R/(alpha_1,...,alpha_r) with a vanishing line of a given slope. The main theorem, Theorem A, asserts that if k has positive characteristic and R satisfies the explicit axioms (C), (SCE), and (F), then for every lambda<1 there is such a complex whose slopes are non-decreasing, strictly below lambda, and of the form k/(k+1), whose final cofibre has a quantised vanishing line, and whose endomorphisms are non-nilpotent; an E3 assumption gives a coherent cube version. Section 3 interprets stable homology as a Bousfield localisation L^f_lambda, and Theorem B derives properties of the localisation from the existence of an admissible Smith–Toda complex, including quantisation of vanishing lines and Adams periodicity. Sections 8 and 9 introduce the stability Hopf algebra Delta_R and argue that it controls the possible stabilisation maps. The paper also contains a characteristic-zero counterexample and several extended examples.","tokens_in":71165,"tokens_out":6986,"duration_ms":76149,"significance":"If the main theorem is correct, this is a substantial unifying framework: it converts the problem of proving higher-order homological stability into a construction in a filtered category of modules, and it gives a concrete numerical prediction, the quantisation of stabilisation slopes as k/(k+1), that is sharp by the characteristic-zero counterexample in Section 10.3. The axioms are explicit and the conditional structure is clear, and the paper credits its debts to published work of F. Cohen and to the author's earlier framework [14,15]. The introduction of the stability Hopf algebra and the precise statements in Theorems 8.2 and 9.6 are a genuine conceptual contribution. There are no fitted parameters and no circular use of the target theorem. The main reservation is that a load-bearing algebraic classification in Section 5.1 is asserted rather than proved, so the numerical form of the main theorem is not yet fully verifiable from the manuscript as written.","major_comments":[{"comment":"The slope quantisation in Theorem A depends on the claim that the non-nilpotent free graded-commutative generators X_{*,*,*} of pi_{*,*,*}(Ctau tensor fil_*R) are supported only in tridegrees D=N-1 for p=2, and D=N-1 or D=N-2 with N even for p odd. The text describes this as an elementary but laborious calculation and cites [14, Section 16.2] for a basis of the W_1-algebra, but does not prove the restricted claim about non-nilpotent generators below the diagonal, nor does it give a precise statement from the literature. This classification is used to identify the possible slopes, to define the total order in Lemma 5.3(ii), and hence to run the induction in Theorem 5.6. If a further family of non-nilpotent generators with other slopes exists, then Theorem A(i), the quantisation of vanishing lines in Section 1.5.1, and the numerical form of the stability Hopf algebra slogan would all fail. A full derivation, or an exact reference with the stated result, must be supplied.","section":"Section 5.1, Definition 5.2 and Lemma 5.3"},{"comment":"The E3 variant used in Theorem A(iv) rests on a second asserted classification: for k>=3 the multiplicative generators below the diagonal are claimed to be exactly the admissible iterated Dyer–Lashof operations listed in Lemma 5.9, with the proof again omitted as 'elementary but laborious'. This lemma determines the basis x_i for the E3 version of the construction, and so it is load-bearing for the coherent-cube statement in Theorem A(iv). The manuscript should either prove Lemma 5.9 or state and prove the precise Cohen-basis calculation in an appendix, rather than deferring it to an unpublished companion or to a reference that does not contain the claim.","section":"Section 5.4, Lemma 5.9"},{"comment":"The condition (†) in Theorem 7.1 is stated using the notation H^R_{n,d}(S), which is not defined in the manuscript as presented. Since Theorem 7.1 is the engine behind both the changing-rings results and the comparison with the stability Hopf algebra in Theorem 8.2, the reader cannot verify that the hypothesis is satisfied by the map R -> r in Corollary 8.4. Please define H^R_{n,d}(-) explicitly and indicate where the relative vanishing estimates in Corollary 8.4 are proved, or state them as a lemma without referring only to [14, Theorem 15.9].","section":"Section 7, Theorem 7.1 and Section 8.2"}],"minor_comments":[{"comment":"The indexing of the spectral sequence in (2.4) is nonstandard; the apology is fine, but it would help to give one concrete example of the correspondence between the filtration degree f and the differential index r.","section":"Section 2.7"},{"comment":"The displayed description pi_{*,*}(RB) = k[r,b] tensor Sym^*[generators of slope >= 1/2] is stated as a consequence of F. Cohen's work but without a reference for this exact presentation; a citation to [14, Section 16] or [10] would be useful.","section":"Section 3, Example 3.9"},{"comment":"After Theorem 6.12 the text says that the remaining endomorphisms 'may be omitted' by the argument of Section 6.3; a short final statement packaging the resulting Smith–Toda complex, with its vanishing line and non-nilpotence properties, would make the connection to Theorem A(iii) more direct.","section":"Section 6.3, Theorem 6.12"},{"comment":"The remark honestly records that Theorem 9.1 and Theorem 9.2 are not known without commutativity; this limits Theorem A(iv) to E3-algebras. The paper should state clearly, near Theorem A, that the cube statement is conditional on this currently unproved noncommutative extension if the E3 assumption is relaxed.","section":"Remark 9.3"}],"recommendation":"major_revision","confidential_remarks":"The central framework is plausible and the paper is ambitious and well written, but I would not accept it in its current form because the numerical content of Theorem A depends on an omitted classification in Section 5.1. This is fixable within the scope of the paper: the author presumably has the calculation and can include it in an appendix. I also recommend asking for a definition of H^R in Theorem 7.1. I do not see evidence of a fatal flaw in the main induction apart from the missing generator classification."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe short version: this is a real and substantial paper. Randal-Williams organizes higher-order homological stability along chromatic lines, proves a general existence theorem (Theorem A) for E2-algebras over positive-characteristic fields satisfying (C), (SCE), and (F), and introduces a stability Hopf algebra that is meant to govern which stabilisation maps occur. If the main theorem holds, it recovers known higher-order examples and adds a genuinely new organizing principle. This deserves an ordinary peer review, not a desk reject.\n\nWhat is genuinely new is the theorem itself and the framing. Previous higher-order stability results were proved family by family; here existence of higher stabilisation maps is deduced from axioms. The Bousfield-localisation reading of stable homology, the quantisation of vanishing lines, and the characteristic-zero counterexample (credited to Burklund) all earn their place. The paper is honest about what it does not prove: the SCE axiom is stated as a hypothesis, and the E3 version in Theorem A(iv) is explicitly conditional.\n\nThe central argument appears sound to me. The elaborate filtered-object and spectral-sequence machinery is presented with enough care that a patient reader can follow the induction in Section 5. The main soft spot is exactly what the stress-test note identifies: the classification in Section 5.1 of non-nilpotent multiplicative generators below the diagonal is load-bearing. The claim that the free graded-commutative generators in tridegrees d<n are given by the free restricted λ1-algebra V is described as an “elementary but laborious calculation” but no calculation or precise reference is supplied. Lemma 5.3 proves slope restrictions for V, but the identification of the generators with V is asserted. Since Theorem A(i) and the quantisation of vanishing lines depend on this, a referee should ask for the argument to be written out or for a complete reference to [14]. I do not think it is wrong; it is under-supported.\n\nTwo further, milder caveats. The paper leans heavily on imported results from [14] and F. Cohen's computation of free E2-algebra homology; that is normal in this area, but it means full verification is beyond any single referee session. And SCE is a real hypothesis, not a consequence of the other axioms; the paper says so, which is to its credit, but it means the general theorem is conditional in an essential way.\n\nWho is this for? Algebraic topologists working on homological stability, group homology, or chromatic phenomena. The companion paper will presumably be important too. My recommendation: send to peer review, with referees asked to expand Section 5.1 and to check the technical imports from [14].","headline":"A serious theory-building paper that gives a general higher-order homological stability theorem in positive characteristic; referee it, but ask for the Section 5.1 generator classification to be expanded.","tokens_in":71764,"tokens_out":3010,"would_cite":true,"duration_ms":31285,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55P42","16T05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Over a field of positive characteristic, every graded E2-algebra satisfying three standard axioms admits higher-order stabilisation maps at every slope below 1, with slopes quantised to k/(k+1) and governed entirely by its stability Hopf…","keywords":["homological stability","higher-order stabilisation","chromatic homotopy theory","Smith–Toda complexes","E2-algebras","Bousfield localisation","stability Hopf algebra","vanishing lines"],"falsifier":"Build an $E_2$-algebra over a field of positive characteristic satisfying (C), (SCE), and (F) together with a finite module that has a vanishing line of slope $3/5$ but no vanishing line of slope $1/2$: the quantisation theorem (Theorems A and B) forces every effective slope below 1 to be exactly $k/(k+1)$, so exhibiting a genuinely different critical slope would refute the central claim.","tokens_in":70663,"feed_emoji":"🎨","tokens_out":16986,"duration_ms":145964,"temperature":0.7,"pith_summary":"This paper organises every kind of homological stability into one theory, in the same way that the chromatic perspective organises stable homotopy theory. For a graded $E_2$-algebra $\\mathbf{R}$ (the homologies of a sequence of spaces or groups, assembled into one object) over a field of positive characteristic, it proves that if three standard axioms hold — connectedness, the standard connectivity estimate, and finite type — then for every slope $\\lambda < 1$ there are higher stabilisation maps $\\alpha_i$, each defined only after killing the previous ones, whose successive cofibres eventually have a vanishing line of slope $\\lambda$, with all slopes quantised to the fractions $k/(k+1)$. It then identifies the object that decides which slopes can occur: the stability Hopf algebra $\\Delta_{\\mathbf{R}}$, built from the diagonal homotopy of the bar construction $k \\otimes_{\\mathbf{R}} k$, whose cohomology has exactly the same stability patterns as $\\mathbf{R}$. The payoff is a definition of stable homology in the higher stable range as a Bousfield localisation (a universal approximation killing a prescribed class of small modules), leading to a chromatic tower, monochromatic layers, and periodic families generated by the higher stabilisation maps.","feed_headline":"One Hopf algebra dictates every stability slope","feed_subtitle":"Higher-order stabilisation maps are organised like a chromatic tower; slopes below 1 are forced to k/(k+1).","key_machinery":"The central objects are the Smith–Toda complexes themselves: iterated cofibres $\\mathbf{R}/(\\alpha_1,\\dots,\\alpha_r)$ of endomorphisms in which each map $\\alpha_i$ is only definable after its predecessors have been killed, and the construction is ordered by the slope $d_i/n_i$ of its bidegree. The existence proof works in the category of filtered $\\mathbf{R}$-modules: the canonical multiplicative filtration $\\mathrm{fil}_*\\mathbf{R}$ has associated graded a free $E_2$-algebra, whose free generators below the diagonal form a totally ordered list $x_1, x_2, \\dots$ (built from Dyer–Lashof operations on the diagonal classes), and the theorem produces for each $x_i$ an $x_i$ self-map $\\phi_i$, detected on the associated graded by a $p$-th power that survives to infinity in the spectral sequence of an endomorphism object — the analogue of the Periodicity Theorem for these filtered objects, with the connectivity estimate (SCE) supplying the differential-killing vanishing. The organising object is the stability Hopf algebra $\\Delta_{\\mathbf{R}}$, the diagonal truncation of the $E_1$-bialgebra $k \\otimes_{\\mathbf{R}} k$; the comparison map $\\mathbf{R} \\to \\mathrm{Cobar}(\\Delta_{\\mathbf{R}})$ satisfies the hypothesis (†) of a change-of-rings theorem that lets nilpotence of endomorphisms be detected after this base change, reducing questions about which Smith–Toda complexes exist and are efficient to the cohomology of a connected graded Hopf algebra.","core_discovery":"The central claim is Theorem A: that higher-order homological stability is not a collection of special cases but a structural consequence of the axioms. For $\\mathbf{R} \\in \\mathrm{Alg}_{E_2}(D(k)^{\\mathbb{Z}})$ over a field $k$ of positive characteristic, satisfying connectedness (C), the standard connectivity estimate (SCE), namely $\\pi_{n,d}(k \\otimes_{\\mathbf{R}} k) = 0$ for $d < n$, and finite type (F), there is for any $\\lambda < 1$ a sequence of $\\mathbf{R}$-module endomorphisms $\\alpha_i : \\mathbf{R}/(\\alpha_1,\\dots,\\alpha_{i-1}) \\otimes S_{n_i,d_i} \\to \\mathbf{R}/(\\alpha_1,\\dots,\\alpha_{i-1})$ whose cofibres, the Smith–Toda complexes $\\mathbf{R}/(\\alpha_1,\\dots,\\alpha_i)$, have non-decreasing slopes $d_i/n_i$ strictly below $\\lambda$ and of the form $k/(k+1)$, the last one carrying a vanishing line of slope $\\lambda$; positive characteristic is essential, since the analogous statement with non-nilpotent endomorphisms fails over $\\mathbb{Q}$ (Section 10.3). The paper further claims that the stability Hopf algebra $\\Delta_{\\mathbf{R}} = \\bigoplus_{n \\geq 0} \\pi_{n,n}(k \\otimes_{\\mathbf{R}} k)$ completely governs these patterns: the map $\\mathbf{R} \\to \\mathrm{Cobar}(\\Delta_{\\mathbf{R}})$ satisfies a change-of-rings theorem and a restricted nilpotence theorem, so base-change preserves and detects non-nilpotent endomorphisms of positive slope, and the stability theory of $\\mathbf{R}$ below slope $1$ becomes exactly the cohomology of a connected graded Hopf algebra. In the higher stable range, stable homology is proposed to be the finite Bousfield localisation $L^f_\\lambda$, which is smashing, quantised by the same slopes, and decomposes into monochromatic layers carrying Adams periodicity.","pith_inferences":["A direct empirical test of the framework: compute $\\Delta_{\\mathbf{R}}$ for families of groups where secondary stability is unknown, such as general linear groups over rings with nontrivial lower $K$-theory or automorphism groups of free nilpotent groups; the framework predicts that the secondary stabilisation slopes are exactly the slopes of the indecomposables of the coinvariants of the $E_1$-St","The quantisation theorem suggests a retrofitting principle for the literature: any existing stability range whose slope is not of the form $k/(k+1)$ should be read as a lower bound only, with the next quantised slope above it as the natural target for an improved range; surveying old stability theorems would show how many ranges this principle would upgrade.","The framework implies that stable homology beyond slope 0 need not be connective — the red-blue configuration example already acquires classes in negative homological degree after localisation — so future computations of higher stable homology should expect torsion and nontrivial extensions rather than a connective answer."],"forward_implications":["Every $E_2$-algebra satisfying (C), (SCE), and (F) over a positive-characteristic field has higher-order stabilisation maps for every slope below 1, so its stability range is always of the quantised form $\\frac{k}{k+1}n + \\kappa$; any proven range with a different slope is therefore not optimal.","Stable homology in the $\\lambda$-range is the smashing Bousfield localisation $L^f_\\lambda$, and since quantised slopes give identical functors there are only countably many distinct stable homologies, assembling into a chromatic tower $M \\to \\cdots \\to L^f_{3/4}(M) \\to L^f_{2/3}(M) \\to L^f_{1/2}(M)$.","Each monochromatic layer carries Adams periodicity: the endomorphisms of slope $\\frac{k-1}{k}$ in the layer induce periodicity isomorphisms in a band of degrees, producing infinite periodic families analogous to the $\\alpha,\\beta,\\gamma$ families of stable homotopy theory.","The stability Hopf algebra $\\Delta_{\\mathbf{R}}$ (the coinvariants of the $E_1$-Steinberg modules in group examples) completely determines which stabilisation maps exist, so discovering new stability theorems reduces to computing the cohomology of a connected graded Hopf algebra.","The regime is sharp: the same conclusions fail over $\\mathbb{Q}$ when non-nilpotence is required (Section 10.3), and with the stated axioms no vanishing line of slope $\\geq 1$ can be forced (Section 10.2)."],"supporting_citations":[{"why":"Supplies the cellular $E_k$-algebra framework: the canonical multiplicative filtration, the indecomposables formula, the splitting complex used to verify (SCE), and the reference for Cohen's free-algebra description.","marker":"[14]"},{"why":"Supplies the description of the homology of free $E_2$-algebras from which the totally ordered generators $x_i$ below the diagonal are derived.","marker":"[10]"},{"why":"Supplies the Periodicity Theorem whose proof the construction of Smith–Toda complexes is modelled on, including the centrality argument making self-maps unique up to $p$-th powers.","marker":"[25]"},{"why":"Supplies the construction of finite Bousfield localisation that Section 3 follows in defining the functors $L^f_\\lambda$.","marker":"[42]"},{"why":"Supplies the secondary homological stability theorem for surface mapping class groups, the motivating example the framework is designed to explain.","marker":"[15]"},{"why":"Supplies Adams' periodicity theorem in homological algebra, the namesake for the periodicity in monochromatic layers and a case later shown to be an instance of the framework.","marker":"[2]"},{"why":"Supplies the filtered-object and $\\tau$-Bockstein machinery (filtered homotopy groups, uniform $\\tau$-torsion) used to construct and compare self-maps.","marker":"[9]"},{"why":"Supplies the finiteness and collapse theorems for Cotor of finite commutative Hopf algebras used in the simultaneous-stabilisation method of Section 9.","marker":"[54]"}],"fun_headline_variants":["Stability slopes dictated by a single Hopf algebra","Chromatic tower organizes higher-order stability","One Hopf algebra encodes all stability slopes","Higher-order stability from one chromatic Hopf algebra"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole construction rests on axiom (SCE), the standard connectivity estimate that the bar construction $k \\otimes_{\\mathbf{R}} k$ has no homotopy in bidegrees below the diagonal; it holds for the group examples through the high connectivity of their splitting complexes, but it is not implied by the other axioms and fails for some $E_2$-algebras, and without it the vanishing estimates and the construction of Smith–Toda complexes collapse.","fun_headline_variants_meta":{"raw":{"variants":["Stability slopes dictated by a single Hopf algebra","Chromatic tower organizes higher-order stability","One Hopf algebra encodes all stability slopes","Higher-order stability from one chromatic Hopf algebra"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000654,"raw_usage":{"total_tokens":3154,"prompt_tokens":1260,"completion_tokens":1894,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":876,"completion_tokens_details":{"reasoning_tokens":1837}},"tokens_in":876,"tokens_out":1894,"duration_ms":13357,"temperature":1.0,"reasoning_tokens":1837,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:42:30.036338+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Build an $E_2$-algebra over a field of positive characteristic satisfying (C), (SCE), and (F) together with a finite module that has a vanishing line of slope $3/5$ but no vanishing line of slope $1/2$: the quantisation theorem (Theorems A and B) forces every effective slope below 1 to be exactly $k/(k+1)$, so exhibiting a genuinely different critical slope would refute the central claim.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the description of the homology of free $E_2$-algebras from which the totally ordered generators $x_i$ below the diagonal are derived."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Periodicity Theorem whose proof the construction of Smith–Toda complexes is modelled on, including the centrality argument making self-maps unique up to $p$-th powers."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the construction of finite Bousfield localisation that Section 3 follows in defining the functors $L^f_\\lambda$."},{"cited_title":"Galatius, A","cited_arxiv_id":null,"evidence_quote":"Supplies the secondary homological stability theorem for surface mapping class groups, the motivating example the framework is designed to explain."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Adams' periodicity theorem in homological algebra, the namesake for the periodicity in monochromatic layers and a case later shown to be an instance of the framework."},{"cited_title":"Burklund, J","cited_arxiv_id":null,"evidence_quote":"Supplies the filtered-object and $\\tau$-Bockstein machinery (filtered homotopy groups, uniform $\\tau$-torsion) used to construct and compare self-maps."},{"cited_title":"Wilkerson","cited_arxiv_id":null,"evidence_quote":"Supplies the finiteness and collapse theorems for Cotor of finite commutative Hopf algebras used in the simultaneous-stabilisation method of Section 9."}],"review_version":2}