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arxiv: 2501.11295 · v2 · pith:QADRYKI4new · submitted 2025-01-20 · 🧮 math.CO · math.AG· math.AT

Filtrations of Tope Spaces of Oriented Matroids

Kris Shaw , Chi Ho Yuen This is my paper

Pith reviewed 2026-05-23 05:37 UTC · model grok-4.3

classification 🧮 math.CO math.AGmath.AT
keywords oriented matroidstope spacesfiltrationsVarchenko-Gelfand degree filtrationKalinin spectral sequenceQuillen augmentation filtrationsign cosheaf
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The pith

Three filtrations of the tope space of an oriented matroid coincide over Z/2Z.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper compares three filtrations of the tope space of an oriented matroid: the dual Varchenko-Gelfand degree filtration, the filtration from Kalinin's spectral sequence, and the one from Quillen's augmentation filtration. It proves that these filtrations and the maps between them coincide when the coefficients are Z/2Z. It also shows that the dual Varchenko-Gelfand degree filtration can be turned into a filtration of the Z-sign cosheaf on the fan of the underlying matroid. If this holds, different constructions yield the same filtered space over finite fields of characteristic two, so properties transfer between them.

Core claim

The central claim is that the dual Varchenko-Gelfand degree filtration, the filtration from Kalinin's spectral sequence, and the filtration from Quillen's augmentation filtration of the tope space of an oriented matroid coincide over Z/2Z together with the maps between them. The paper further shows that the dual Varchenko-Gelfand degree filtration can be made into a filtration of the Z-sign cosheaf on the fan of the underlying matroid.

What carries the argument

The coincidence of the three filtrations of the tope space over Z/2Z coefficients, which unifies the constructions.

If this is right

  • The three filtrations become equivalent over Z/2Z so any one can stand in for the others.
  • The dual Varchenko-Gelfand degree filtration works with Z coefficients when applied to the sign cosheaf.
  • The maps between filtrations are compatible with reducing coefficients from Z to Z/2Z.
  • Constructions previously limited to Z/2Z coefficients now extend via this identification.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This result might make it easier to compute invariants by picking the filtration that is simplest for a given matroid.
  • Similar identifications could exist for other rings or for related objects in matroid theory.
  • The extension to the sign cosheaf may enable new calculations in the topology of matroid fans.

Load-bearing premise

The three filtrations are defined on the same tope space of an arbitrary oriented matroid and the maps between them are well-defined and compatible with the coefficient ring change from Z to Z/2Z.

What would settle it

An oriented matroid where the three filtrations or their maps do not coincide when computed with Z/2Z coefficients would disprove the claim.

Figures

Figures reproduced from arXiv: 2501.11295 by Chi Ho Yuen, Kris Shaw.

Figure 1
Figure 1. Figure 1: A real hyperplane arrangement realising U2,3. When an oriented matroid is realised by a real hyperplane arrangement consisting of hyperplanes indexed by E, the covectors are precisely the sign vectors encoding the relative position of the points in the ambient space (see Example 2.1): we fix a positive halfspace for the i-th hyperplane, and record which side a point is in (including the case of being on th… view at source ↗
Figure 2
Figure 2. Figure 2: The six 2-dimensional cells of the Salvetti complex from Example 2.6 2.3. The Orlik–Solomon Algebra and the Cordovil Algebra. Next we in￾troduce the Orlik–Solomon algebra of a matroid. Its significance is that when the matroid is representable over C, its Orlik–Solomon algebra is naturally isomorphic to the (de Rham) cohomology ring of the complement of any complex hyperplane arrangement representing the m… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript compares three filtrations of the tope space of an oriented matroid: the dual Varchenko-Gelfand degree filtration, the filtration arising from Kalinin's spectral sequence, and the filtration from Quillen's augmentation. It proves that the filtrations and the maps between them coincide over Z/2Z. It further shows that the dual Varchenko-Gelfand degree filtration extends to a filtration of the Z-sign cosheaf on the fan of the underlying matroid, extending prior Z/2Z work with applications to patchworking in real algebraic geometry.

Significance. If the identifications hold, the result unifies three distinct constructions of filtrations on tope spaces, providing a common framework that strengthens the link between combinatorial topology and real algebraic geometry. The extension from Z/2Z to Z coefficients for the cosheaf filtration is a concrete technical advance over the earlier work cited in the abstract.

minor comments (2)
  1. The introduction would benefit from a short diagram or table explicitly listing the three filtrations, their coefficient rings, and the maps whose coincidence is claimed, to make the statement of the main theorem easier to parse at first reading.
  2. Notation for the tope space and the sign cosheaf is introduced without a dedicated preliminary subsection; adding one would help readers who are not already expert in oriented matroid cohomology.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the clear summary of its contributions, and the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines the dual Varchenko-Gelfand, Kalinin, and Quillen filtrations independently on the tope space of an arbitrary oriented matroid, then constructs explicit maps to show coincidence over Z/2Z and extends the first to a Z-sign cosheaf filtration on the underlying matroid fan. No equations, fitted parameters, or self-definitional reductions appear in the abstract or described claims. The reference to prior work by the first author (with Renaudineau) concerns only the Z/2Z case and is not invoked as a uniqueness theorem or load-bearing premise for the new Z-coefficient results. The derivation chain remains self-contained against external definitions of the filtrations and does not reduce any central claim to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Pure mathematical comparison of filtrations; no numerical fitting or new postulated objects. Relies on standard definitions of oriented matroids, tope spaces, and the three named filtrations.

axioms (2)
  • domain assumption Standard properties of oriented matroids and their tope spaces as topological objects
    Invoked implicitly when defining the filtrations on the tope space.
  • domain assumption The three filtrations are defined on the same underlying space and admit natural maps between them
    Required for the statement that the filtrations and maps coincide.

pith-pipeline@v0.9.0 · 5655 in / 1434 out tokens · 24917 ms · 2026-05-23T05:37:29.175509+00:00 · methodology

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Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages

  1. [1]

    Betti numbers of real semistable degenerations via real logarithmic geometry

    Emiliano Ambrosi and Matilde Manzaroli. Betti numbers of real semistable degenerations via real logarithmic geometry. arXiv:2211.12134, 2022

    work page arXiv 2022

  2. [2]

    Federico Ardila and Caroline J. Klivans. The Bergman complex of a matroid and phylogenetic trees. J. Combin. Theory Ser. B , 96(1):38–49, 2006

    work page 2006

  3. [3]

    The homology and shellability of matroids and geometric lattices

    Anders Bj¨ orner. The homology and shellability of matroids and geometric lattices. InMatroid applications, volume 40 of Encyclopedia Math. Appl., pages 226–283. Cambridge Univ. Press, Cambridge, 1992

    work page 1992

  4. [4]

    Anders Bj¨ orner, Michel Las Vergnas, Bernd Sturmfels, Neil White, and G¨ unter M. Ziegler. Oriented matroids, volume 46 of Encyclopedia of Mathematics and its Applications . Cam- bridge University Press, Cambridge, second edition, 1999

    work page 1999

  5. [5]

    Anders Bj¨ orner and G¨ unter M. Ziegler. Combinatorial stratification of complex arrangements. J. Amer. Math. Soc. , 5(1):105–149, 1992

    work page 1992

  6. [6]

    Cordovil

    R. Cordovil. A commutative algebra for oriented matroids. volume 27, pages 73–84. 2002. Geometric combinatorics (San Francisco, CA/Davis, CA, 2000)

    work page 2002

  7. [7]

    Topological properties of real algebraic varieties: du cot´ e de chez rokhlin.Russian Mathematical Surveys , 55(4):735–814, aug 2000

    A I Degtyarev and V M Kharlamov. Topological properties of real algebraic varieties: du cot´ e de chez rokhlin.Russian Mathematical Surveys , 55(4):735–814, aug 2000

    work page 2000

  8. [8]

    The Orlik-Solomon complex and Milnor fibre homology

    Graham Denham. The Orlik-Solomon complex and Milnor fibre homology. volume 118, pages 45–63. 2002. Arrangements in Boston: a Conference on Hyperplane Arrangements (1999)

    work page 2002

  9. [9]

    Equivariant cohomology and conditional oriented matroids

    Galen Dorpalen-Barry, Nicholas Proudfoot, and Jidong Wang. Equivariant cohomology and conditional oriented matroids. Int. Math. Res. Not. IMRN , (11):9292–9322, 2024

    work page 2024

  10. [10]

    Oriented matroids

    Jon Folkman and Jim Lawrence. Oriented matroids. J. Combin. Theory Ser. B , 25(2):199– 236, 1978. 30 KRIS SHA W AND CHI HO YUEN

    work page 1978

  11. [11]

    On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions, and orientations of graphs

    Curtis Greene and Thomas Zaslavsky. On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions, and orientations of graphs. Trans. Amer. Math. Soc., 280(1):97–126, 1983

    work page 1983

  12. [12]

    Tropical homology

    Ilia Itenberg, Ludmil Katzarkov, Grigory Mikhalkin, and Ilia Zharkov. Tropical homology. Math. Ann., 374(1-2):963–1006, 2019

    work page 2019

  13. [13]

    Cohomology of real algebraic varieties

    IO Kalinin. Cohomology of real algebraic varieties. Journal of Mathematical Sciences , 131(1):5323–5344, 2005

    work page 2005

  14. [14]

    On matroids and Orlik-Solomon algebras

    Yukihito Kawahara. On matroids and Orlik-Solomon algebras. Ann. Comb., 8(1):63–80, 2004

    work page 2004

  15. [15]

    Equivariant cohomology and the Varchenko-Gelfand filtration

    Daniel Moseley. Equivariant cohomology and the Varchenko-Gelfand filtration. J. Algebra, 472:95–114, 2017

    work page 2017

  16. [16]

    Combinatorics and topology of complements of hyperplanes

    Peter Orlik and Louis Solomon. Combinatorics and topology of complements of hyperplanes. Invent. Math., 56(2):167–189, 1980

    work page 1980

  17. [17]

    Arrangements of hyperplanes , volume 300 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]

    Peter Orlik and Hiroaki Terao. Arrangements of hyperplanes , volume 300 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Springer-Verlag, Berlin, 1992

    work page 1992

  18. [18]

    D. Quillen. On the associated graded ring of a group ring. J. Algebra, 10:411–418, 1968

    work page 1968

  19. [19]

    Real phase structures on matroid fans and matroid orientations

    Johannes Rau, Arthur Renaudineau, and Kris Shaw. Real phase structures on matroid fans and matroid orientations. Journal of the London Mathematical Society , 106(4):3687–3710, 2022

    work page 2022

  20. [20]

    Real phase structures on tropical man- ifolds and patchworks in higher codimension

    Johannes Rau, Arthur Renaudineau, and Kris Shaw. Real phase structures on tropical man- ifolds and patchworks in higher codimension. arXiv:2310.08313, 2023

    work page arXiv 2023

  21. [21]

    Bounding the Betti numbers of real hypersurfaces near the tropical limit

    Arthur Renaudineau and Kris Shaw. Bounding the Betti numbers of real hypersurfaces near the tropical limit. Ann. Sci. ´Ec. Norm. Sup´ er. (4), 56(3):945–980, 2023

    work page 2023

  22. [22]

    Salvetti

    M. Salvetti. Topology of the complement of real hyperplanes inCN. Invent. Math., 88(3):603– 618, 1987

    work page 1987

  23. [23]

    A. N. Varchenko and I. M. Gelfand. Heaviside functions of a configuration of hyperplanes. Funktsional. Anal. i Prilozhen. , 21(4):1–18, 96, 1987

    work page 1987

  24. [24]

    Bilinear form of real configuration of hyperplanes

    Alexandre Varchenko. Bilinear form of real configuration of hyperplanes. Adv. Math. , 97(1):110–144, 1993

    work page 1993

  25. [25]

    Hilbert’s sixteenth problem

    George Wilson. Hilbert’s sixteenth problem. Topology, 17(1):53–73, 1978

    work page 1978

  26. [26]

    The Orlik-Solomon algebra and the Bergman fan of a matroid

    Ilia Zharkov. The Orlik-Solomon algebra and the Bergman fan of a matroid. J. G¨ okova Geom. Topol. GGT, 7:25–31, 2013. Department of Mathematics, University of Oslo, Norway Email address: krisshaw@math.uio.no Department of Applied Mathematics, National Yang Ming Chiao Tung Univer- sity, Taiwan Email address: chyuen@math.nctu.edu.tw

    work page 2013