Filtrations of Tope Spaces of Oriented Matroids

Technical Summary: Filtrations of Tope Spaces of Oriented Matroids

Problem Statement
The paper investigates the structure of the tope space of an oriented matroid $M$, defined as the free module generated by the set of topes (maximal covectors) of $M$. Specifically, the authors address the relationship between three distinct filtrations of this space when considered over the field $\mathbb{Z}/2\mathbb{Z}$. These filtrations arise from different mathematical contexts:

1. The Dual Varchenko–Gelfand filtration ($P_\bullet$), derived from the degree filtration of the ring of Heaviside functions on the tope space.
2. The Kalinin filtration ($K_\bullet$), induced by the spectral sequence of the Salvetti complex equipped with a conjugation involution.
3. The Quillen filtration ($Q_\bullet$), constructed via the augmentation filtration of group algebras associated with affine subspaces of topes adjacent to complete flags of flats.

While these filtrations were known to exist, their precise relationship and the compatibility of their associated maps to homology groups remained to be fully established, particularly regarding whether they coincide and how they relate to the Orlik–Solomon algebra and the Salvetti complex homology.

Methodology
The authors employ a combination of combinatorial algebra, spectral sequence theory, and the topology of Salvetti complexes.

• Combinatorial Descriptions: The paper utilizes "prefix chains" (also known as bricks) to explicitly describe the generators of the dual Varchenko–Gelfand filtration. It also uses affine coordinate subspaces to describe the Quillen filtration.
• Topological Models: The Salvetti complex ($Sal_M$) serves as the topological model for the "complexification" of the oriented matroid. The authors analyze the involution (conjugation) on this complex, noting that its fixed locus corresponds to the set of topes.
• Spectral Sequences: The Kalinin filtration is analyzed via the spectral sequence associated with the involution on the Salvetti complex. The authors prove this sequence degenerates at the first page for oriented matroids, simplifying the filtration structure.
• Algebraic Isomorphisms: The authors utilize the Björner–Ziegler isomorphism ($BZ^\vee$) between the homology of the Salvetti complex and the dual of the Orlik–Solomon algebra. They also introduce the Cordovil algebra as a commutative analogue to the Orlik–Solomon algebra that better captures oriented matroid data over $\mathbb{Z}$.

Key Results

1. Coincidence of Filtrations (Theorem A):
   The primary result establishes that for any oriented matroid $M$, the three filtrations of the $\mathbb{Z}/2\mathbb{Z}$-tope space are identical:
   $$Q_\bullet(M) = P_\bullet(M) = K_\bullet(M)$$
   The proof proceeds by establishing a chain of inclusions $Q_\bullet \subseteq P_\bullet \subseteq K_\bullet$ and demonstrating that the dimensions of the associated graded pieces match for all three, forcing equality.

2. Compatibility of Associated Maps (Theorem B):
   The paper compares the maps induced by these filtrations to the homology of the Salvetti complex $H_p(Sal_M; \mathbb{Z}/2\mathbb{Z})$ and the dual Orlik–Solomon algebra $OS^p(M; \mathbb{Z}/2\mathbb{Z})$.

   • The map $bvp$ (from the Kalinin filtration/Viro homomorphism) and the map $ap$ (from the dual Varchenko–Gelfand filtration/Denham isomorphism) are shown to be equal.
   • The diagram involving the Quillen map $qp$ and the dual Björner–Ziegler isomorphism $BZ^\vee$ is proven to commute.
   • Consequently, all three filtrations descend to the same isomorphism between their associated graded pieces and the homology groups (up to the identification provided by $BZ^\vee$).

3. Extension to Integral Coefficients and Cosheaves (Theorem C):
   While the Quillen and Kalinin filtrations do not behave well over $\mathbb{Z}$ (the Quillen filtration does not stabilize, and a naive Kalinin extension is ill-defined), the dual Varchenko–Gelfand filtration is well-defined over $\mathbb{Z}$.

   • The authors prove that this filtration is functorial with respect to initial matroids (deletion/contraction operations).
   • This allows the construction of a filtration of the $\mathbb{Z}$-sign cosheaf on the polyhedral fan of the underlying matroid.
   • The associated graded pieces of this integral filtration are identified with the Cordovil cosheaves ($A_p$), leading to a short exact sequence of cosheaves:
     $$0 \to P_{p+1} \to P_p \to A_p \to 0$$
   • The paper notes that while the mod 2 reduction of the Cordovil algebra is isomorphic to the Orlik–Solomon algebra, a direct lift of the Orlik–Solomon algebra to $\mathbb{Z}$ in this exact sequence is impossible (demonstrated via Example 6.17), necessitating the use of the Cordovil algebra for the integral case.

Significance and Claims
The paper claims to unify three previously distinct approaches to filtering the tope space of an oriented matroid. By proving their coincidence over $\mathbb{Z}/2\mathbb{Z}$, the authors provide a robust combinatorial framework that connects:

• Singularity Theory: Via the Varchenko–Gelfand degree filtration and its relation to asymptotic filtrations.
• Real Algebraic Geometry: Via the Kalinin filtration and its connection to the Smith–Thom inequality and Viro's patchworking.
• Combinatorial Homology: Via the Quillen filtration and its relation to the Orlik–Solomon algebra.

The extension to $\mathbb{Z}$-coefficients via the Cordovil algebra is presented as a necessary refinement for studying integral homology in the context of tropical manifolds with real phase structures. The authors suggest that this integral filtration of the sign cosheaf provides a tropical/combinatorial version of Kalinin's filtration, offering a tool to compute the $\mathbb{Z}$-homology of real parts of hypersurfaces arising from patchworking. The work relies on and extends previous results by Renaudineau, the first author, and others, specifically generalizing the mod 2 results to an integral setting where the Orlik–Solomon algebra is insufficient.