The double of a simplicial complex

Bridget Schreiner; Kathryn Lesh; Nathalie Wahl

The double of a simplicial complex

T0 review · 0 major / 7 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Doubling a simplicial complex preserves weak Cohen-Macaulayness with an explicit dimension bound.

desk verdict 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. read the letter →

arxiv 2506.10436 v1 pith:KRKK7HTM submitted 2025-06-12 math.CO math.AT

Kathryn Lesh , Bridget Schreiner , Nathalie Wahl This is my paper · ORCID

classification math.COmath.AT MSC 05E4555U10

keywords simplicialcomplexesweaklyCohen-Macaulayr-tuplingmatchingconnectivityhomologicalstabilitydestabilizationsimplexwiseinjectivemaps

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 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.

What carries the argument

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.

What would settle it

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 $\lfloor (n-3)/3\rfloor - 1$.

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Extended reading notes

Core claim

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 $r/(k(r+1))$.

Load-bearing premise

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.

Editorial extensions

If this is right

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 $1/2$.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

What 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.

The 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.

A 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.

Who 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.

Referee Report

0 major / 7 minor

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.

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.

minor comments (7)

[§3, proof of Theorem A, link dimension] 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.
[§3, step 1 of proof of Theorem A] 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.
[§3, step 2 of proof of Theorem A] 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.
[§2, matching complex identification] 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.
[§2, definition of wCM] 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.
[§4, Proposition 4.6] 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.
[Introduction and §3] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem A is derived from external matching-complex connectivity results and independent cited lemmas.

full rationale

The derivation chain is self-contained with respect to the target claim. Theorem A is proved by a direct connectivity argument: for a map S^k to D_r(X), the paper first lifts to X_{(k+2)r} using [9, Lem 3.8] and [7, Thm 2.4], then extends across each simplex using the known connectivity of D_r(alpha_m) for alpha_m a simplex, quoted from Athanasiadis [1] via Theorem 2.2. The base case D_r(Delta_n) is imported as an external result, not derived from Theorem A; the general case does not assume Theorem A for the original X. The link reduction uses Lemma 3.2 and applies the theorem to the link, which has strictly smaller dimension, so it is a legitimate induction rather than a circular appeal. The citations [9] and [13] include one of the present authors (Wahl), but they are published results with independent proofs and are used as black boxes with stated hypotheses; this is legitimate external support, not self-citation load-bearing. There are no fitted parameters, no quantity is renamed as a prediction, and no uniqueness claim is imported from the authors' prior work. The only noted internal issue is an arithmetic inequality in Step 1, and the paper verifies a stronger inequality than a corrected dimension would require, so it does not affect the conclusion. Thus the central claim has independent mathematical content and no circularity is present.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central theorem is proved using established results on matching complexes and connectivity of poset subcomplexes; no free parameters are fitted. The construction D_r(X) is a new definition but not an independent postulate.

assumptions (4)

standard math Theorem 2.2 (Athanasiadis): Dr(Δn) is wCM of dimension floor((n+1-r)/(r+1)).
Used in the proof of Theorem A (Section 3, step 2) to extend maps across simplices. This is an external established result.
standard math Lemma 3.8 of Hatcher-Wahl [9]: If X is wCM of dimension n, then Xm (simplices of dimension at least m-1) is (n-m)-connected.
Used in Lemma 3.1 and in the proof of Theorem A to get null-homotopy of f-hat. External published result.
standard math Theorem 2.4 of Galatius-Randal-Williams [7]: In a wCM complex, maps from spheres extend to disks simplexwise injectively.
Used to ensure simplexwise injective extension in the proof of Theorem A. External published result.
domain assumption Assumption that the monoidal category (C, ⊕, 0) is locally standard at (A, X) and that S_n(A,X) is wCM.
Section 4, used to relate destabilization complexes to r-tupling. Holds for standard examples.

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Pith. "Pith review of The double of a simplicial complex." pith.science (2026). https://pith.science/paper/KRKK7HTM

@misc{pith2026250610436,
  author       = {Pith},
  title        = {Pith review of: The double of a simplicial complex},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KRKK7HTM}},
  note         = {Machine review of arXiv:2506.10436}
}

read the original abstract

We introduce the notion of doubling and r-tupling for simplicial complexes, a notion reminiscent to that of matching complexes in graph theory. We prove a connectivity result for such complexes and relate r-tupling to stabilizing r times faster in homological stability.

Figures

Figures reproduced from arXiv: 2506.10436 by the authors.

**Figure 1.** Vertex of D(∆2 ) and edge of D(∆3 ) 2. Doubling, r-tupling, and matching complexes Recall that a simplicial complex X = (X0,P) is the data of a set of vertices X0 together with a collection P of subsets of X0 including all the singletons and closed under taking subsets. A p-simplex of X is then a subset τ = {x0, . . . , xp} ∈ P of cardinality p + 1. We denote by Xp the collection of p-simplices of X. Definition 2.1.… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[1]

Athanasiadis

Christos A. Athanasiadis. Decompositions and connectivity of matching and chessboard complexes. Dis- crete Comput. Geom., 31(3):395–403, 2004

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[2]

Bj¨ orner, L

A. Bj¨ orner, L. Lov´ asz, S. T. Vre´ cica, and R. T.ˇZivaljevi´ c. Chessboard complexes and matching complexes. J. London Math. Soc. (2), 49(1):25–39, 1994

work page 1994
[3]

Shellable and Cohen-Macaulay partially ordered sets

Anders Bj¨ orner. Shellable and Cohen-Macaulay partially ordered sets. Trans. Amer. Math. Soc., 260(1):159–183, 1980

work page 1980
[4]

S. Bouc. Homologie de certains ensembles de 2-sous-groupes des groupes sym´ etriques. J. Algebra, 150(1):158–186, 1992

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[5]

Fluch, Marco Marschler, Stefan Witzel, and Matthew C

Kai-Uwe Bux, Martin G. Fluch, Marco Marschler, Stefan Witzel, and Matthew C. B. Zaremsky. The braided Thompson’s groups are of type F∞. J. Reine Angew. Math., 718:59–101, 2016. With an appendix by Zaremsky

work page 2016
[6]

Frank D. Farmer. Cellular homology for posets. Math. Japon., 23(6):607–613, 1978/79

work page 1978
[7]

Homological stability for moduli spaces of high dimensional manifolds

Sø ren Galatius and Oscar Randal-Williams. Homological stability for moduli spaces of high dimensional manifolds. I. J. Amer. Math. Soc., 31(1):215–264, 2018

work page 2018
[8]

Cellular Ek-algebras, 2023

Soren Galatius, Alexander Kupers, and Oscar Randal-Williams. Cellular Ek-algebras, 2023. To appear in Asterisque

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[9]

Stabilization for mapping class groups of 3-manifolds

Allen Hatcher and Nathalie Wahl. Stabilization for mapping class groups of 3-manifolds. Duke Math. J., 155(2):205–269, 2010

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Homological stability of topological moduli spaces

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Uniform twisted homological stability, 2025

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Athanasiadis

Christos A. Athanasiadis. Decompositions and connectivity of matching and chessboard complexes. Dis- crete Comput. Geom., 31(3):395–403, 2004

work page 2004

[2] [2]

Bj¨ orner, L

A. Bj¨ orner, L. Lov´ asz, S. T. Vre´ cica, and R. T.ˇZivaljevi´ c. Chessboard complexes and matching complexes. J. London Math. Soc. (2), 49(1):25–39, 1994

work page 1994

[3] [3]

Shellable and Cohen-Macaulay partially ordered sets

Anders Bj¨ orner. Shellable and Cohen-Macaulay partially ordered sets. Trans. Amer. Math. Soc., 260(1):159–183, 1980

work page 1980

[4] [4]

S. Bouc. Homologie de certains ensembles de 2-sous-groupes des groupes sym´ etriques. J. Algebra, 150(1):158–186, 1992

work page 1992

[5] [5]

Fluch, Marco Marschler, Stefan Witzel, and Matthew C

Kai-Uwe Bux, Martin G. Fluch, Marco Marschler, Stefan Witzel, and Matthew C. B. Zaremsky. The braided Thompson’s groups are of type F∞. J. Reine Angew. Math., 718:59–101, 2016. With an appendix by Zaremsky

work page 2016

[6] [6]

Frank D. Farmer. Cellular homology for posets. Math. Japon., 23(6):607–613, 1978/79

work page 1978

[7] [7]

Homological stability for moduli spaces of high dimensional manifolds

Sø ren Galatius and Oscar Randal-Williams. Homological stability for moduli spaces of high dimensional manifolds. I. J. Amer. Math. Soc., 31(1):215–264, 2018

work page 2018

[8] [8]

Cellular Ek-algebras, 2023

Soren Galatius, Alexander Kupers, and Oscar Randal-Williams. Cellular Ek-algebras, 2023. To appear in Asterisque

work page 2023