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Filtrations of Tope Spaces of Oriented Matroids

Kris Shaw, Chi Ho Yuen

TL;DR

This work analyzes three filtrations on the tope space of an oriented matroid—the dual Varchenko–Gelfand degree filtration, Kalinin’s spectral-sequence filtration, and Quillen’s augmentation filtration—and proves they coincide modulo 2, with compatible maps to Salvetti-homology. It then extends the dual VG filtration to a filtration of the $\mathbb{Z}$-sign cosheaf on the matroid fan, relating the associated graded pieces to Cordovil algebras and enabling a cosheaf-theoretic perspective that connects to patchworking in real algebraic geometry. The paper also clarifies the interplay between these filtrations and the initial-matroid structure, showing functoriality and providing explicit identifications via prefix chains and NBC bases, culminating in a projectivization that ties to tropical homology via sign cosheaves. Together, these results unify combinatorial, topological, and tropical approaches to the tope space and its filtrations, with implications for real-algebraic-geometric constructions such as patchworking. The findings offer a cohesive framework linking Orlik–Solomon and Cordovil algebras, Salvetti complexes, and matroid fans, and open pathways to compute and compare homological invariants in a purely combinatorial setting.

Abstract

We compare three filtrations of the tope space of an oriented matroid. The first is the dual Varchenko-Gelfand degree filtration, the second filtration is from Kalinin's spectral sequence, and the last one derives from Quillen's augmentation filtration. We show that all three filtrations and the respective maps coincide over $\mathbb{Z}/ 2\mathbb{Z}$. We also show that the dual Varchenko-Gelfand degree filtration can be made into a filtration of the $\mathbb{Z}$-sign cosheaf on the fan of the underlying matroid. This was previously carried out with $\mathbb{Z}/ 2\mathbb{Z}$-coefficients by the first author and Renaudineau using the Quillen filtration and has applications to real algebraic geometry via patchworking.

Filtrations of Tope Spaces of Oriented Matroids

TL;DR

This work analyzes three filtrations on the tope space of an oriented matroid—the dual Varchenko–Gelfand degree filtration, Kalinin’s spectral-sequence filtration, and Quillen’s augmentation filtration—and proves they coincide modulo 2, with compatible maps to Salvetti-homology. It then extends the dual VG filtration to a filtration of the -sign cosheaf on the matroid fan, relating the associated graded pieces to Cordovil algebras and enabling a cosheaf-theoretic perspective that connects to patchworking in real algebraic geometry. The paper also clarifies the interplay between these filtrations and the initial-matroid structure, showing functoriality and providing explicit identifications via prefix chains and NBC bases, culminating in a projectivization that ties to tropical homology via sign cosheaves. Together, these results unify combinatorial, topological, and tropical approaches to the tope space and its filtrations, with implications for real-algebraic-geometric constructions such as patchworking. The findings offer a cohesive framework linking Orlik–Solomon and Cordovil algebras, Salvetti complexes, and matroid fans, and open pathways to compute and compare homological invariants in a purely combinatorial setting.

Abstract

We compare three filtrations of the tope space of an oriented matroid. The first is the dual Varchenko-Gelfand degree filtration, the second filtration is from Kalinin's spectral sequence, and the last one derives from Quillen's augmentation filtration. We show that all three filtrations and the respective maps coincide over . We also show that the dual Varchenko-Gelfand degree filtration can be made into a filtration of the -sign cosheaf on the fan of the underlying matroid. This was previously carried out with -coefficients by the first author and Renaudineau using the Quillen filtration and has applications to real algebraic geometry via patchworking.
Paper Structure (18 sections, 38 theorems, 56 equations, 2 figures)

This paper contains 18 sections, 38 theorems, 56 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Oriented Matroids
  4. Salvetti Complexes
  5. The Orlik--Solomon Algebra and the Cordovil Algebra
  6. Matroid fans and cosheaves
  7. Three filtrations of the tope space
  8. Dual Varchenko--Gelfand filtration
  9. Kalinin filtration
  10. Quillen filtration
  11. Comparison of the filtrations
  12. The dual VG filtration contains the Quillen filtration
  13. The Kalinin filtration contains the dual VG filtration
  14. Proof of Theorem \ref{['mainthm:maps']}
  15. A ${\mathbb Z}$-coefficients filtration of sign cosheaf on a matroid fan
  16. ...and 3 more sections

Key Result

Theorem 1

Let $M$ be an oriented matroid. The filtrations of the $\mathbb{Z}/2\mathbb{Z}$--tope space ${\mathcal{Q}}_{\bullet}(M), {\mathcal{K}}_{\bullet}(M),$ and $\overline{{\mathcal{P}}}_{\bullet}(M)$ all coincide.

Figures (2)

  • Figure 1: A real hyperplane arrangement realising $U_{2,3}$.
  • Figure 2: The six 2-dimensional cells of the Salvetti complex from Example \ref{['ex:SalvettiU23']}

Theorems & Definitions (90)

  • Theorem 1
  • Theorem 2
  • Proposition 1.1
  • Theorem 3
  • Example 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Example 2.6
  • ...and 80 more