{"id":"7a8ab0ee-538d-4b39-beda-431647756712","arxiv_id":"2506.10436","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The r-tupling of a weakly Cohen-Macaulay simplicial complex is again weakly Cohen-Macaulay, with dimension roughly (n-r+1)/(r+1).","lead":"This paper introduces a new construction for simplicial complexes, called r-tupling, and proves that it preserves a strong connectivity property. The result generalizes known theorems about matching complexes and provides new bounds for homological stability under fast stabilization.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified.","rationale":"The reader identifies Theorem 2.2, the shellability of Athanasiadis, as the weakest assumption. I do not see this as a load-bearing weak point: it is a published, standard result, and the paper's restatement of it is accurate. The more delicate part is the internal inductive proof of Theorem A. I checked the connectivity inequalities in Lemma 3.1 and in Steps 1 and 2 of the proof. The joins and connectivity bounds are consistent under the paper's conventions, including the treatment of empty links as (−2)-connected. The final inequality for links, (n − r(p+1) − 1 − 2r)/(r+1) ≥ (n − r + 1)/(r+1) − p − 2, is valid. The use of [7, Thm 2.4] to obtain simplexwise injective extensions is appropriate because the relevant subcomplexes are wCM of sufficiently high dimension. The main theorem therefore appears correct, and the minor dimension overstatement in Step 1 is harmless because the proof's verified inequality already exceeds the requirement for the corrected dimension. I recommend leaving the reader's ACCEPT verdict unchanged.","tokens_in":11618,"tokens_out":46290,"duration_ms":544989,"concrete_test":"Recompute the dimension of the target complex in Step 1 by applying Lemma 3.1 to Y = Link_X(δf(σ0)) rather than to X; the corrected dimension is n − r(k + 2) + 1, and the proof's inequality m − 1 ≤ n − r(k + 2) − 1 is still sufficient for a simplexwise injective extension. This confirms that the stated larger dimension in the paper is unnecessary and not load-bearing.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. I read the central argument of Theorem A carefully, including the connectivity proof in Section 3, the use of Lemma 3.1 in Step 1, the extension argument in Step 2, and the application of Theorem 2.2. The external input on shellability of matching complexes is standard and well-supported, and the internal induction on skeleta appears sound: the sphere map into Dr(X) is first lifted to X_{r(k+2)}, then extended to a disc via [9, Lem 3.8], and finally projected back using the connectivity of Dr(α_m). The link argument via Lemma 3.2 correctly reduces the wCM condition for links to the same theorem applied to smaller complexes. I found one place where the stated dimension of a middle complex in Step 1 appears too large, but the inequality actually verified in the proof is stronger than what a corrected dimension requires, so this does not threaten the conclusion.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a construction called the r-tupling D_r(X) of a simplicial complex X, whose vertices are the (r−1)-simplices of X and whose p-simplices are collections of r-(r−1)-simplices whose union is a simplex of X of dimension (p+1)r−1. The main result, Theorem A, states that if X is weakly Cohen-Macaulay (wCM) of dimension n, then D_r(X) is wCM of dimension floor((n−r+1)/(r+1)). The proof proceeds by a connectivity argument through the intermediate complex X_{(k+2)r} of the barycentric subdivision, using known results on matching complexes (Athanasiadis), a lemma of Hatcher–Wahl, and a theorem of Galatius–Randal-Williams. Section 4 applies the result to the destabilization complexes of Randal-Williams–Wahl, showing that fast stabilization by r at a time has connectivity governed by the r-tupling of the underlying simplicial complex.","tokens_in":11755,"tokens_out":24155,"duration_ms":238893,"significance":"If Theorem A is correct, it gives a natural construction that preserves weak Cohen-Macaulayness with an explicit connectivity bound, generalizing known results for matching complexes of complete hypergraphs. The homological stability application is interesting and gives a new perspective on fast stabilization. The paper is concise and the main proof is transparent, with a clear separation of the external inputs. The result should be of interest to both combinatorialists and topologists and fits the journal's scope. The proof is not machine-checked, but the arguments are standard and reproducible.","major_comments":[],"minor_comments":[{"comment":"In the paragraph after Lemma 3.2, the claim that Link_X δτ is wCM of dimension n − r(p + 1) − 1 should read n − r(p + 1): since δτ is a simplex of dimension r(p+1)−1, the link of a q-dimensional simplex in a wCM complex of dimension n is wCM of dimension n − q − 1. The subsequent connectivity inequality is unaffected, but the displayed formula should be corrected.","section":"§3, proof of Theorem A, link dimension"},{"comment":"The dimension of the complex Link_X(δf(σ0))_{r(k+2)−r(p′+1)} is stated as n − r(k+2) + r(p′+1) + 1; applying Lemma 3.1 to the link of δf(σ0), which has dimension n − r(p′+1), gives n − r(k+2) + 1. The extension condition m − 1 ≤ n − r(k+2) is still satisfied under the standing assumption, so the argument remains valid once the formula is corrected.","section":"§3, step 1 of proof of Theorem A"},{"comment":"In the induction step for the extension g-hat, the image complex is written as X_{2k+4}; for general r this should be X_{(k+2)r}. This appears to be a typo, as the surrounding text uses the correct bound.","section":"§3, step 2 of proof of Theorem A"},{"comment":"The sentence 'Let K_{n+1} be the complete graph on n vertices' should say 'on n+1 vertices', so that the isomorphism D(Δ_n) ≅ M(K_{n+1}) is consistent.","section":"§2, matching complex identification"},{"comment":"The phrase 'if only the connectivity condition for the links is satisfied' is misleading, since the standard definition of weakly Cohen-Macaulay also requires the complex itself to be (n−1)-connected. The equivalent characterization via the n-skeleton that follows is the correct statement; the initial phrasing should be revised to avoid ambiguity.","section":"§2, definition of wCM"},{"comment":"The dimension formulas in Proposition 4.6 appear to assume that (n+k−a)/k is an integer. If it is not, floors should be inserted in the input dimension and propagated through the calculation of the output connectivity slope. Please clarify this point.","section":"§4, Proposition 4.6"},{"comment":"There are several small typos: 'see e.g.; [13] or [17]' has an extra semicolon, and in the proof of Theorem A, 'δf(s0)' should be 'δf(σ0)' in the factorization of f-hat on ∂σ.","section":"Introduction and §3"}],"recommendation":"minor_revision","confidential_remarks":"I support publication after minor revisions. The proof of Theorem A is sound; the issues are local typos and imprecise dimension statements that do not affect the validity of the argument. The paper's reliance on results co-authored by one of the present authors is appropriate, as those results are established and used as black boxes. The manuscript fits the journal's scope and should be suitable for publication once the minor corrections are made."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this paper gives a new construction, the r-tupling of a simplicial complex, and proves a genuine generalization of known matching-complex connectivity results. Theorem A says that if X is weakly Cohen–Macaulay of dimension n, then D_r(X) is wCM of dimension floor((n-r+1)/(r+1)). The proof cleanly adapts the Hatcher–Wahl method, using the known simplex case as the external input. The application to fast stabilization is also new, and the authors are honest that the stability slopes they get are not optimal.\n\nWhat is genuinely good: the construction is natural, the statement is clean, and the paper is well-written. The proof uses standard tools: Lemma 3.1 promotes Hatcher–Wahl's connectivity result to full wCM, Lemma 3.2 identifies links of the r-tupling with r-tuplings of links, and then the induction on skeleta goes through with the simplex case as the base input. The reliance on established external results—Athanasiadis, Hatcher–Wahl, Galatius–Randal-Williams—is appropriate, and there is no circularity. The identification of D(Δ^n) with matching complexes of complete r-hypergraphs is clearly explained.\n\nThe soft spots are minor. The dimension formula in Step 1 of the proof of Theorem A appears to have a typo: the paper claims the relevant link complex is wCM of dimension n - r(k+2) + r(p'+1) + 1, but the correct computation gives n - r(k+2) + 1 (the r(p'+1) term cancels). The inequality used in the extension argument is still satisfied—in fact with more room than the authors need—so the conclusion is unaffected. Worth fixing before publication.\n\nA second point, not a flaw: the paper positions itself as a note, and the homological-stability consequences are explicitly non-optimal. That is fine; the value here is the new construction and the general theorem, not the sharpest stability ranges.\n\nWho this is for: combinatorial topologists and anyone working on homological stability via simplicial complexes. It is a useful, quotable tool. The paper deserves serious refereeing; I would send it out and likely accept after the typo is corrected.","headline":"A clean new construction and a genuine generalization of matching-complex connectivity; the proof is sound, with only a minor typo in a dimension formula.","tokens_in":12286,"tokens_out":4690,"would_cite":true,"duration_ms":45873,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E45","55U10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Doubling a simplicial complex preserves weak Cohen-Macaulayness with an explicit dimension bound.","keywords":["simplicial complexes","weakly Cohen-Macaulay","r-tupling","matching complexes","connectivity","homological stability","destabilization complexes","simplexwise injective maps"],"falsifier":"Compute the connectivity of $D_r(X)$ for a weakly Cohen-Macaulay complex X that is not a simplex, for instance a triangulation of the 4-sphere with r = 2; Theorem A predicts $D_2(X)$ is connected, so finding a nonzero $H_0$ would disprove it. A more targeted check is to compute the link of a vertex in $D_2(X)$ for a small weakly Cohen-Macaulay X and compare its connectivity with the predicted value $\\lfloor (n-3)/3\\rfloor - 1$.","tokens_in":2076,"feed_emoji":"🔺","tokens_out":2559,"duration_ms":126145,"temperature":0.7,"pith_summary":"This paper introduces a construction, the r-tupling of a simplicial complex: the vertices are the (r−1)-simplices of a given complex X, and higher simplices are disjoint collections whose union is again a simplex of X. The main theorem asserts that if X is weakly Cohen-Macaulay of dimension n, then its r-tupling $D_r(X)$ is weakly Cohen-Macaulay of dimension $\\lfloor (n-r+1)/(r+1)\\rfloor$. This matters because it upgrades a single known connectivity statement about simplexes into a general statement about all weakly Cohen-Macaulay complexes, and because the r-tupling of a simplex is exactly the matching complex of a complete hypergraph, so the theorem subsumes a family of matching-complex results. The paper also shows that the construction explains what happens to destabilization complexes when homological stabilization is taken r steps at a time.","feed_headline":"Doubling a simplicial complex preserves weak Cohen-Macaulayness","feed_subtitle":"The r-tupling of any weakly Cohen-Macaulay complex is again weakly Cohen-Macaulay, with matching complexes as a special case.","key_machinery":"The r-tupling $D_r(X)$ is the central object: its vertices are the $(r-1)$-simplices of X, and its p-simplices are collections $\\{\\tau_0,\\ldots,\\tau_p\\}$ whose union is a $((p+1)r-1)$-simplex of X. Two structural facts carry the argument. First, Lemma 3.2 gives the link formula $\\operatorname{Link}_{D_r(X)}(\\tau) \\cong D_r(\\operatorname{Link}_X(\\delta\\tau))$, so the weakly Cohen-Macaulay property for $D_r(X)$ follows once the connectivity statement is known for complexes of the form $D_r(Y)$ with Y weakly Cohen-Macaulay. Second, the proof uses the subposet $X_{(k+2)r}$ of the barycentric subdivision consisting of simplices with at least $(k+2)r$ vertices; a lemma of Hatcher and Wahl says this subposet is highly connected, and Lemma 3.1 of the paper upgrades that to weakly Cohen-Macaulay, allowing simplexwise-injective extensions via a theorem of Galatius and Randal-Williams. The base case $D_r(\\Delta^m)$ is the matching complex of the complete r-hypergraph, whose shellability was established by Athanasiadis.","core_discovery":"The paper's central result is Theorem A: for every weakly Cohen-Macaulay simplicial complex X of dimension n, the r-tupling $D_r(X)$ is weakly Cohen-Macaulay of dimension $\\lfloor (n-r+1)/(r+1)\\rfloor$. The proof reduces the question to the known case where X is a single simplex: because a p-simplex $\\tau$ of $D_r(X)$ has an underlying simplex $\\delta(\\tau)$ of X, the link of $\\tau$ in $D_r(X)$ is $D_r$ of the link of $\\delta(\\tau)$ in X; that link is again weakly Cohen-Macaulay, so the full link condition follows once the connectivity statement is known for the complex itself. The connectivity statement is proved by mapping a sphere into $D_r(X)$, lifting the map to the subcomplex $X_{(k+2)r}$ of the barycentric subdivision, extending that lift over a disc using the weak Cohen-Macaulay hypothesis, and then projecting back simplex by simplex with the help of the known weak Cohen-Macaulay property of $D_r(\\Delta^m)$. The paper additionally proves Proposition 4.6, which says that under the standard assumptions of homological stability, if the destabilization complexes $W_n(A,X)$ are highly connected, then the fast-stabilization complexes $W_n(A,X^{\\oplus r})$ are weakly Cohen-Macaulay with an explicit connectivity slope $r/(k(r+1))$.","pith_inferences":["The two-step induction is likely transplantable: the same lifting argument into $X_{(k+2)r}$ and projection back through $D_r(\\Delta^m)$ should work whenever a base complex with the needed weak Cohen-Macaulay property is available, so replacing the simplex by another small family of seed complexes would yield a general class of such theorems.","The paper does not address whether the dimension bound $\\lfloor (n-r+1)/(r+1)\\rfloor$ is sharp for arbitrary weakly Cohen-Macaulay X; testing the double of joins or suspensions of simplexes could reveal whether the bound is tight beyond the matching-complex case.","The link formula suggests a representation-theoretic by-product: when a group acts on X by permuting simplices, the induced action on the first non-vanishing homology of $D_r(X)$ could be studied as a functor of the action on X, extending the analysis already carried out for symmetric groups."],"forward_implications":["Every weakly Cohen-Macaulay complex of dimension n produces, for every r, a new weakly Cohen-Macaulay complex $D_r(X)$ of the explicit dimension $\\lfloor (n-r+1)/(r+1)\\rfloor$.","When X is the n-simplex, $D_r(X)$ is the matching complex of the complete r-hypergraph on $n+1$ vertices, so Theorem A recovers the known connectivity bounds for matching complexes as a special case.","The link formula reduces the whole weakly Cohen-Macaulay property of $D_r(X)$ to connectivity of the complex itself, so any future improvement of the connectivity bound for weakly Cohen-Macaulay X automatically improves the bound for all links.","Under the local standardness assumptions used in homological stability, the fast-stabilization complexes $W_n(A,X^{\\oplus r})$ are weakly Cohen-Macaulay with connectivity slope $r/(k(r+1))$; for symmetric groups the double case is sharp and the resulting stability slope is the optimal $1/2$."],"supporting_citations":[{"why":"Supplies the base case: the r-tupling of a simplex, identified with the complete r-hypergraph matching complex, is weakly Cohen-Macaulay with the dimension used in Theorem A.","marker":"[1]"},{"why":"Provides the earlier weak Cohen-Macaulay bound for hypergraph matching complexes that Theorem A generalizes.","marker":"[2]"},{"why":"Used to pass from shellability of the relevant skeleton to the weakly Cohen-Macaulay property in Theorem 2.2.","marker":"[3]"},{"why":"Provides the simplexwise-injective extension theorem used in step 1 of the proof to extend maps over discs without vertex collisions.","marker":"[7]"},{"why":"Supplies Lemma 3.8 on the connectivity of subposets $X_m$ and Proposition 3.5 on complete join complexes, both used in the main proof and in Section 4.","marker":"[9]"},{"why":"Establishes the destabilization-complex framework and the theorem translating connectivity of $W_n$ into homological stability for automorphism groups.","marker":"[13]"},{"why":"Shows the matching-complex bound is sharp for r = 2, which the paper cites to show that the symmetric-group destabilization bounds cannot be improved.","marker":"[14]"}],"fun_headline_variants":["r-tupling preserves weak Cohen-Macaulayness in complexes","Doubling a complex: weak Cohen-Macaulayness persists","New connectivity result for doubled simplicial complexes","Weak Cohen-Macaulayness survives complex doubling","r-tupling keeps complexes weakly Cohen-Macaulay"],"cache_read_input_tokens":14592,"weakest_assumption_plain":"The proof depends on the known theorem that the r-tupling of a simplex is weakly Cohen-Macaulay with exactly the stated dimension; if that base case were false or weaker, the lifting argument would not deliver the claimed bound for arbitrary X.","fun_headline_variants_meta":{"raw":{"variants":["r-tupling preserves weak Cohen-Macaulayness in complexes","Doubling a complex: weak Cohen-Macaulayness persists","New connectivity result for doubled simplicial complexes","Weak Cohen-Macaulayness survives complex doubling","r-tupling keeps complexes weakly Cohen-Macaulay"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001303,"raw_usage":{"total_tokens":5286,"prompt_tokens":888,"completion_tokens":4398,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":504,"completion_tokens_details":{"reasoning_tokens":4314}},"tokens_in":504,"tokens_out":4398,"duration_ms":32944,"temperature":1.0,"reasoning_tokens":4314,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:27:09.282998+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the connectivity of $D_r(X)$ for a weakly Cohen-Macaulay complex X that is not a simplex, for instance a triangulation of the 4-sphere with r = 2; Theorem A predicts $D_2(X)$ is connected, so finding a nonzero $H_0$ would disprove it. A more targeted check is to compute the link of a vertex in $D_2(X)$ for a small weakly Cohen-Macaulay X and compare its connectivity with the predicted value $\\lfloor (n-3)/3\\rfloor - 1$.","supporting_citations":[{"cited_title":"Athanasiadis","cited_arxiv_id":null,"evidence_quote":"Supplies the base case: the r-tupling of a simplex, identified with the complete r-hypergraph matching complex, is weakly Cohen-Macaulay with the dimension used in Theorem A."},{"cited_title":"Bj¨ orner, L","cited_arxiv_id":null,"evidence_quote":"Provides the earlier weak Cohen-Macaulay bound for hypergraph matching complexes that Theorem A generalizes."},{"cited_title":"Shellable and Cohen-Macaulay partially ordered sets","cited_arxiv_id":null,"evidence_quote":"Used to pass from shellability of the relevant skeleton to the weakly Cohen-Macaulay property in Theorem 2.2."},{"cited_title":"Homological stability for moduli spaces of high dimensional manifolds","cited_arxiv_id":null,"evidence_quote":"Provides the simplexwise-injective extension theorem used in step 1 of the proof to extend maps over discs without vertex collisions."},{"cited_title":"Stabilization for mapping class groups of 3-manifolds","cited_arxiv_id":null,"evidence_quote":"Supplies Lemma 3.8 on the connectivity of subposets $X_m$ and Proposition 3.5 on complete join complexes, both used in the main proof and in Section 4."},{"cited_title":"Homological stability for automorphism groups","cited_arxiv_id":null,"evidence_quote":"Establishes the destabilization-complex framework and the theorem translating connectivity of $W_n$ into homological stability for automorphism groups."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows the matching-complex bound is sharp for r = 2, which the paper cites to show that the symmetric-group destabilization bounds cannot be improved."}],"review_version":1}