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.
| Comments: | 38 pages, 7 figures, comments welcome |
| Subjects: | Combinatorics (math.CO); Commutative Algebra (math.AC); Algebraic Geometry (math.AG) |
| Cite as: | arXiv:2605.24695 [math.CO] |
| (or arXiv:2605.24695v1 [math.CO] for this version) | |
| https://doi.org/10.48550/arXiv.2605.24695 arXiv-issued DOI via DataCite (pending registration) |
From: Juliette Bruce [view email]
[v1]
Sat, 23 May 2026 18:18:04 UTC (74 KB)
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