On the connection between coordinate and diagonal arrangement complements

Vsevolod Tril

On the connection between coordinate and diagonal arrangement complements

T0 review · 0 major / 3 minor · reviewed 2026-05-23 · grok-4.3

Challenge this review Download PDF Read on arXiv ↗

Pith's one-line read For simplicial complexes where any two missing faces share a vertex, the coordinate arrangement complement U(K) is the double suspension of the diagonal arrangement complement D(K).

desk verdict The paper proves that U(K) is the double suspension of D(K) when K has the property that any two missing faces share a vertex. read the letter →

arxiv 2409.18001 v1 submitted 2024-09-26 math.AT math.CO

Vsevolod Tril This is my paper

classification math.ATmath.CO

keywords simplicialcomplexesarrangementcomplementscoordinatearrangementsdiagonalsuspensionshomotopytypecomplexspacereal

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

The paper examines diagonal arrangement complements in complex space. It restricts attention to simplicial complexes in which every pair of missing faces intersects in at least one vertex. Under this restriction the author shows that the complement of the coordinate arrangement is the double suspension of the complement of the diagonal arrangement. The corresponding statement in real space replaces the double suspension by a single suspension.

What carries the argument

The restricted class of simplicial complexes K where missing faces pairwise share a vertex, which allows the proof that U(K) is the double suspension of D(K).

What would settle it

A simplicial complex K violating the common-vertex condition for missing faces, together with an explicit computation showing that U(K) is not the double suspension of D(K).

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Extended reading notes

Core claim

We consider the class of simplicial complexes K in which any two missing faces have a common vertex, and prove that the coordinate arrangement complement U(K) is the double suspension of the diagonal arrangement complement D(K). In the case of subspace arrangements in R^m the coordinate arrangement complement U_R(K) is the single suspension of D_R(K).

Load-bearing premise

The condition that any two missing faces of K share a common vertex is required for the suspension relation to hold.

Editorial extensions

If this is right

U(K) and D(K) are related by suspension, so their homotopy groups differ by a shift.
The real and complex cases differ by one suspension level.
This relation holds specifically when missing faces share vertices.
Topology of one complement determines the other up to suspension.

Reading between the lines

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

This relation might simplify computations of topological invariants for one arrangement using the other.
It could be tested whether the vertex-sharing condition is necessary or if the relation holds more generally.
Extensions to other types of arrangements or higher-dimensional analogs might follow similar suspension patterns.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The central claim is a homotopy identification: for simplicial complexes K where every pair of missing faces shares a vertex, the coordinate arrangement complement U(K) is homotopy equivalent to the double suspension of the diagonal arrangement complement D(K). The real case gives a single suspension. This is presented as a new statement rather than a restatement of prior work on these spaces. The shared-vertex condition is stated up front as the setting for the result, which keeps the scope clear. The paper does a straightforward job defining the two complements and stating the theorem without extra machinery. That kind of direct relation can be useful for anyone who already computes with one construction and wants a quick way to transfer information to the other. The restriction on K is the main limitation. It excludes many standard examples where missing faces can be vertex-disjoint, so the result covers a proper subclass rather than the general case. The abstract gives no proof outline or key lemmas, which means the actual argument has to be checked in the body for gaps or extra hidden conditions. No circularity or parameter fitting appears in the claim itself. This is for readers already working with arrangement complements or combinatorial topology who need concrete equivalences for specific K. A generalist or someone looking for broad new techniques will not get much out of it. The work is precise enough on its own terms to deserve referee time so the proof can be verified.

Referee Report

0 major / 3 minor

Summary. The manuscript studies diagonal arrangement complements D(K) in C^m (and their real analogues). It restricts to the class of simplicial complexes K in which any two missing faces share a common vertex, and proves that the coordinate arrangement complement U(K) is homotopy equivalent to the double suspension of D(K); the real case yields a single suspension.

Significance. If the stated identification holds, the result supplies an explicit homotopy relation between two families of arrangement complements under a transparent combinatorial hypothesis on K. This may allow transfer of known computations or invariants from one setting to the other and clarifies the effect of the shared-vertex condition on the topology of the complements.

minor comments (3)

[Abstract] The abstract states the result but does not record the precise homotopy equivalence symbol (e.g., ≃ or ≃_h); adding it would improve immediate readability.
A single concrete example of a complex K satisfying the shared-vertex condition (together with the resulting spaces U(K) and D(K)) would help readers verify the scope of the theorem.
Notation for the real and complex versions (U_R, D_R, etc.) should be introduced once and used consistently throughout the text.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript and for recognizing its potential significance in relating two families of arrangement complements under the shared-vertex condition on K. The recommendation of minor revision is noted, but the report lists no specific major comments or requested changes.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The paper states a direct topological theorem: for the explicitly restricted class of simplicial complexes K in which any two missing faces share a vertex, it proves U(K) ≃ Σ² D(K) (and the real analogue U_R(K) ≃ Σ D_R(K)). The abstract and reader's summary present this as a proved identification within a defined combinatorial setting rather than a reduction of any quantity to a fitted parameter, self-citation chain, or definitional renaming. No equations or steps in the provided material reduce the claimed result to its own inputs by construction, and the restriction is foregrounded as the theorem's scope. The derivation is therefore self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The result rests on standard facts about suspensions and arrangement complements in algebraic topology together with the combinatorial restriction on K. No free parameters or invented entities are introduced in the abstract.

assumptions (2)

standard math Standard properties of suspension in homotopy theory (adding two cones) preserve the relevant topological invariants in the expected way.
Invoked implicitly when claiming that U(K) equals the suspension of D(K).
domain assumption The definition of coordinate and diagonal arrangement complements via the simplicial complex K.
The constructions U(K) and D(K) are taken as given from prior literature in the field.

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Cite this review

Pith. "Pith review of On the connection between coordinate and diagonal arrangement complements." pith.science (2026). https://pith.science/paper/2409.18001

@misc{pith2026240918001,
  author       = {Pith},
  title        = {Pith review of: On the connection between coordinate and diagonal arrangement complements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2409.18001}},
  note         = {Machine review of arXiv:2409.18001}
}

abstract

We study diagonal arrangement complements $D(K)$ in $\mathbb{C}^m$. We consider the class of simplicial complexes $K$ in which any two missing faces have a common vertex, and prove that the coordinate arrangement complement $U(K)$ is the double suspension of the diagonal arrangement complement $D(K)$. In the case of subspace arrangements in $\mathbb{R}^m$ the coordinate arrangement complement $U_{\mathbb{R}}(K)$ is the single suspension of $D_{\mathbb{R}}(K)$.

Figures

Figures reproduced from arXiv: 2409.18001 by the authors.

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1 ... any two missing faces of a simplicial complex K have a common vertex. Then there is a homotopy equivalence U(K) ≃ Σ²D(K).

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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Bjorner and V

A. Bjorner and V. Welker. The homology of ”k-equal” manifolds and related partition lattices. Advances in Mathematics, 110(2):277–313, 1995

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Bj¨ orner and G

A. Bj¨ orner and G. M. Ziegler. Combinatorial stratification of complex arrangements. J. Am. Math. Soc., 5(1):105–149, 1992

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V. M. Buchstaber and T. E. Panov. Torus actions, combinatorial topology, and homological algebra. Russian Math. Surveys , 55:825–921, 2000

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V. M. Buchstaber and T. E. Panov. Toric topology, volume 204 of Math. Surv. Monogr. Providence, RI: American Mathematical Society (AMS), 2015

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de Longueville and C

M. de Longueville and C. Schultz. The cohomology rings of complements of subspace arrangements. Mathematische Annalen, 319:625–646, 2001

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Dobrinskaya

N. Dobrinskaya. Loops on polyhedral products and diagonal arrangements. arXiv: Algebraic Topol- ogy, 2009

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[9]

Goresky and R

M. Goresky and R. MacPherson. Stratified Morse Theory . Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics. Springer Berlin Heidelberg, 1988

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Grbi´ c, T

J. Grbi´ c, T. Panov, S. Theriault, and Jie Wu. The homotopy types of moment-angle complexes for flag complexes. Trans. Amer. Math. Soc., 368:6663–6682, 2016

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K. Iriye and D. Kishimoto. Fat-wedge filtration and decomposition of polyhedral products. Kyoto Journal of Mathematics , 59(1):1 – 51, 2019

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Peeva, V

I. Peeva, V. Reiner, and V. Welker. Cohomology of real diagonal subspace arrangements via resolu- tions. Compositio Mathematica, 117(1):107–123, 1999. Department of Mathematics and Mechanics, Moscow State University, Russia Email address : vsevolod.tril@math.msu.ru

1999

Pith tools

Reviewed May 23, 2026 · model on record in the stance chip above.

[1] [1]

V. I. Arnol’d. The cohomology ring of the colored braid group. Math. Notes, 5:138–140, 1969

work page 1969

[2] [2]

Bahri, M

A. Bahri, M. Bendersky, F.R. Cohen, and S. Gitler. The polyhedral product functor: A method of decomposition for moment-angle complexes, arrangements and related spaces. Advances in Mathe- matics, 225(3):1634–1668, 2010

work page 2010

[3] [3]

Bjorner and V

A. Bjorner and V. Welker. The homology of ”k-equal” manifolds and related partition lattices. Advances in Mathematics, 110(2):277–313, 1995

work page 1995

[4] [4]

Bj¨ orner and G

A. Bj¨ orner and G. M. Ziegler. Combinatorial stratification of complex arrangements. J. Am. Math. Soc., 5(1):105–149, 1992

work page 1992

[5] [5]

V. M. Buchstaber and T. E. Panov. Torus actions, combinatorial topology, and homological algebra. Russian Math. Surveys , 55:825–921, 2000

work page 2000

[6] [6]

V. M. Buchstaber and T. E. Panov. Toric topology, volume 204 of Math. Surv. Monogr. Providence, RI: American Mathematical Society (AMS), 2015

work page 2015

[7] [7]

de Longueville and C

M. de Longueville and C. Schultz. The cohomology rings of complements of subspace arrangements. Mathematische Annalen, 319:625–646, 2001

work page 2001

[8] [8]

Dobrinskaya

N. Dobrinskaya. Loops on polyhedral products and diagonal arrangements. arXiv: Algebraic Topol- ogy, 2009

work page 2009