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arXiv · 2605.24695

Explorations of Matroid Complexes

Juliette Bruce·Jacob Bucciarelli·Bailee Zacovic
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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.

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Juliette Bruce, Jacob Bucciarelli, Bailee Zacovic. 2026-05-23. Explorations of Matroid Complexes. https://arxiv.org/abs/2605.24695

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