Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W7162434487
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