Explorations of Matroid Complexes

Bailee Zacovic; Jacob Bucciarelli; Juliette Bruce

Matroid classes with orientations form a Hopf algebra under direct sum and restriction-contraction that proves broad acyclicity.

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2026-06-30 12:53 UTC pith:PEXG4NQ6

load-bearing objection They build a Hopf algebra on oriented matroid classes via bicomplexes, prove acyclicity, run homology computations to n=15, and conjecture a description by odd-wheel matroids.

arxiv 2605.24695 v1 pith:PEXG4NQ6 submitted 2026-05-23 math.CO math.ACmath.AG

Explorations of Matroid Complexes

Juliette Bruce , Jacob Bucciarelli , Bailee Zacovic This is my paper

classification math.CO math.ACmath.AG MSC 05B35

keywords matroid complexesHopf algebradeletion contractionacyclicityhomology computationodd wheel matroidsKontsevich graph complexes

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper equips the vector space spanned by oriented matroid classes with deletion and contraction bicomplexes to unify several matroid complex variants. Direct sum and restriction-contraction turn this space into a connected graded Hopf algebra extending Schmitt's matroid Hopf algebra. The dg-algebra structure from this construction is used to prove broad acyclicity results. Computations of the complexes through ground-set size 9 and a quotient up to size 15 detect nontrivial homology, leading to a conjecture involving odd-wheel matroids.

Core claim

We construct deletion and contraction bicomplexes on the vector space spanned by matroid classes equipped with ground-set orientations. We show that direct sum and restriction-contraction make this space into a connected graded Hopf algebra extending Schmitt's matroid Hopf algebra, and use the resulting dg-algebra structure to prove broad acyclicity results. We compute the total, simple, loopless, regular, binary, and ternary matroid complexes through ground-set size 9, and the connected quotient of the simple loopless regular complex through ground-set size 15. These computations detect nontrivial homology and lead to a conjectural description in terms of odd-wheel matroids.

What carries the argument

Deletion and contraction bicomplexes on the vector space of oriented matroid classes that induce a connected graded Hopf algebra structure.

Load-bearing premise

The vector space spanned by matroid classes equipped with ground-set orientations admits well-defined deletion and contraction bicomplexes that correctly organize the several naturally arising variants.

What would settle it

A computation of the homology for the simple loopless regular matroid complex at ground-set size 16 that contains a class not accounted for by odd-wheel matroids would falsify the conjecture.

Watch this falsifier — get emailed when new claim-graph text bears on it.

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

They build a Hopf algebra on oriented matroid classes via bicomplexes, prove acyclicity, run homology computations to n=15, and conjecture a description by odd-wheel matroids.

read the letter

The core contribution is a single framework that puts deletion and contraction bicomplexes on the vector space of oriented matroid classes, equips it with a connected graded Hopf algebra structure extending Schmitt, and uses the dg-algebra to get acyclicity theorems. They then compute the total, simple, loopless, regular, binary, and ternary complexes through size 9 and the connected quotient of the simple loopless regular one through size 15, finding nontrivial homology that motivates the odd-wheel conjecture.

What stands out is the systematic unification of variants that had been treated separately, plus the actual numbers and the conjecture. The computations are concrete and the Hopf-algebra extension looks like a clean move from the cited Schmitt algebra. The acyclicity results are presented as broad, which is the kind of statement that can be checked once the differentials are written down.

The main soft spot is that the conjecture rests on data up to 15; that is enough to spot a pattern but not enough to rule out other families at larger sizes. The bicomplex definitions and the verification that the Hopf axioms hold are the parts that need the closest look in a review, since the abstract only sketches them. No obvious circularity or parameter fitting appears in the stated claims.

This is for people already working in matroid theory or algebraic combinatorics who want organized data and a new algebraic handle on these complexes. A reader outside that niche will not get much unless they already care about Kontsevich-style graph complexes or matroid Hopf algebras.

It deserves a serious referee. The constructions are new, the computations are reproducible in principle, and the conjecture is falsifiable with more machine time. I would send it out.

Referee Report

0 major / 2 minor

Summary. The paper constructs deletion and contraction bicomplexes on the vector space spanned by oriented matroid classes, unifying several variants. It shows that direct sum and restriction-contraction equip this space with the structure of a connected graded Hopf algebra extending Schmitt's matroid Hopf algebra. The resulting dg-algebra structure is used to prove acyclicity results. Explicit computations are given for the total, simple, loopless, regular, binary, and ternary matroid complexes up to ground-set size 9, and for the connected quotient of the simple loopless regular complex up to size 15; these detect nontrivial homology and motivate a conjecture describing it in terms of odd-wheel matroids.

Significance. If the Hopf algebra axioms, bicomplex differentials, and acyclicity proofs hold, the work supplies a new algebraic framework for matroid complexes modeled on Kontsevich graph complexes, together with concrete computational data that supports a precise homological conjecture. The explicit extension of Schmitt's algebra and the machine-checkable nature of the small-n computations are particular strengths.

minor comments (2)

The abstract and introduction would benefit from a brief explicit statement of the ground-set orientation convention used to define the vector space basis, as this choice organizes all subsequent bicomplexes.
In the computational sections, the tables listing homology dimensions for the various complexes would be easier to compare if a uniform row/column labeling convention (e.g., always listing Betti numbers by homological degree) were adopted throughout.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of its contributions to the algebraic framework extending Schmitt's matroid Hopf algebra, and the recommendation for minor revision. The noted strengths in the bicomplex construction, dg-structure, acyclicity results, and computational support for the odd-wheel conjecture are appreciated.

Circularity Check

0 steps flagged

No significant circularity; constructions are independent definitions and verifications

full rationale

The paper defines deletion/contraction bicomplexes on oriented matroid classes as new structures motivated by Kontsevich graph complexes, then verifies that direct sum and restriction-contraction yield a connected graded Hopf algebra extending Schmitt's (external citation). Acyclicity theorems follow directly from the resulting dg-algebra differentials. Explicit computations up to ground-set size 9 (and 15 for one quotient) are performed by enumeration and detect homology, motivating an odd-wheel conjecture. No equation reduces a claimed result to a fitted parameter, self-citation chain, or input by construction; all load-bearing steps are self-contained algebraic verifications or direct computation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

As a pure-mathematics paper, the work rests on standard definitions and axioms of matroid theory and Hopf algebras; no free parameters, invented entities, or ad-hoc axioms are indicated in the abstract.

axioms (1)

standard math Standard axioms and definitions of matroids, orientations, deletion, contraction, and graded Hopf algebras
The constructions and proofs presuppose the usual properties of these objects as established in the literature.

pith-pipeline@v0.9.1-grok · 5665 in / 1505 out tokens · 50783 ms · 2026-06-30T12:53:27.498637+00:00 · methodology

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read the original abstract

Motivated by Kontsevich's graph complexes, this paper gives a systematic study of matroid complexes. We construct deletion and contraction bicomplexes on the vector space spanned by matroid classes equipped with ground-set orientations, organizing the several naturally arising variants into a single unified framework. We show that direct sum and restriction-contraction make this space into a connected graded Hopf algebra extending Schmitt's matroid Hopf algebra, and use the resulting dg-algebra structure to prove broad acyclicity results. We compute the total, simple, loopless, regular, binary, and ternary matroid complexes through ground-set size $9$, and the connected quotient of the simple loopless regular complex through ground-set size $15$. These computations detect nontrivial homology and lead to a conjectural description in terms of odd-wheel matroids.

discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

[1]

Tropicalizations of locally sym- metric varieties, 2025.arXiv:2505.10504

[ABB+25] Eran Assaf, Madeline Brandt, Juliette Bruce, Melody Chan, and Raluca Vlad. Tropicalizations of locally sym- metric varieties, 2025.arXiv:2505.10504. 2 [ABGW00] Tatiana V. Alekseyevskaya, Alexandre V. Borovik, I. M. Gelfand, and Neil White. Matroid homology. InThe Gelfand Mathematical Seminars, 1996–1999, Gelfand Math. Sem., pages 1–13. Birkhäuser...

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33 [BBC+24] Madeline Brandt, Juliette Bruce, Melody Chan, Margarida Melo, Gwyneth Moreland, and Corey Wolfe

arXiv:2301.10108. 33 [BBC+24] Madeline Brandt, Juliette Bruce, Melody Chan, Margarida Melo, Gwyneth Moreland, and Corey Wolfe. On the top-weight rational cohomology ofA g.Geom. Topol., 28(2):497–538,

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Brown, M

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Formal (non)commutative symplectic geometry

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Feynman diagrams and low-dimensional topology

1 [Kon94] Maxim Kontsevich. Feynman diagrams and low-dimensional topology. InFirst European Congress of Mathe- matics, Vol. II (Paris, 1992), volume 120 ofProgr. Math., pages 97–121. Birkhäuser, Basel,

work page 1992
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The 11-loop graph cohomology, 2025.arXiv:2508.13724

1 [Wil25] Thomas Willwacher. The 11-loop graph cohomology, 2025.arXiv:2508.13724. 1 APPENDIXA. COMPUTEDCOMPLEXES The tables in this section record dimensions of chain groups for the regular, binary, ternary, simple, and loopless deletion matroid complexes, together with the simple loopless connected regular matroid complex. 9 0 8 0 0 7 0 0 0 6 0 0 0 0 5 0...

work page arXiv 2025

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Tropicalizations of locally sym- metric varieties, 2025.arXiv:2505.10504

[ABB+25] Eran Assaf, Madeline Brandt, Juliette Bruce, Melody Chan, and Raluca Vlad. Tropicalizations of locally sym- metric varieties, 2025.arXiv:2505.10504. 2 [ABGW00] Tatiana V. Alekseyevskaya, Alexandre V. Borovik, I. M. Gelfand, and Neil White. Matroid homology. InThe Gelfand Mathematical Seminars, 1996–1999, Gelfand Math. Sem., pages 1–13. Birkhäuser...

work page arXiv 2025

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33 [BBC+24] Madeline Brandt, Juliette Bruce, Melody Chan, Margarida Melo, Gwyneth Moreland, and Corey Wolfe

arXiv:2301.10108. 33 [BBC+24] Madeline Brandt, Juliette Bruce, Melody Chan, Margarida Melo, Gwyneth Moreland, and Corey Wolfe. On the top-weight rational cohomology ofA g.Geom. Topol., 28(2):497–538,

work page arXiv

[3] [3]

Brown, M

2, 4, 14, 15 [BCGP24] Francis Brown, Melody Chan, Søren Galatius, and Sam Payne. Hopf algebras in the cohomology ofAg, GLn(Z), and SLn(Z), 2024.arXiv:2405.11528. 4, 34 [BH25] Paul-Hermann Balduf and Simone Hu. The topological form is the pfaffian form, 2025.arXiv:2503.09558. 1 [BHP24] Francis Brown, Simone Hu, and Erik Panzer. Unstable cohomology ofGL 2n(...

work page arXiv 2024

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Formal (non)commutative symplectic geometry

3 34 [Kon93] Maxim Kontsevich. Formal (non)commutative symplectic geometry. InThe Gel’fand Mathematical Seminars, 1990–1992, pages 173–187. Birkhäuser Boston, Boston, MA,

work page 1990

[5] [5]

Feynman diagrams and low-dimensional topology

1 [Kon94] Maxim Kontsevich. Feynman diagrams and low-dimensional topology. InFirst European Congress of Mathe- matics, Vol. II (Paris, 1992), volume 120 ofProgr. Math., pages 97–121. Birkhäuser, Basel,

work page 1992

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The 11-loop graph cohomology, 2025.arXiv:2508.13724

1 [Wil25] Thomas Willwacher. The 11-loop graph cohomology, 2025.arXiv:2508.13724. 1 APPENDIXA. COMPUTEDCOMPLEXES The tables in this section record dimensions of chain groups for the regular, binary, ternary, simple, and loopless deletion matroid complexes, together with the simple loopless connected regular matroid complex. 9 0 8 0 0 7 0 0 0 6 0 0 0 0 5 0...

work page arXiv 2025