Abstract
Let ℳn be a linear hyperplane arrangement in ℝn. We define two corresponding posetsG k(ℳn andV k(ℳn) of oriented matroids, which approximate the GrassmannianG k(ℝn) and the Stiefel manifoldV k(ℝn). The basic conjectures are that the “OM-Grassmannian”G k(ℳn) has the homotopy type ofG k(ℝn), and that the “OM-Stiefel bundle” Δπ: ΔV k(ℳn) → ΔG k(ℳn) is a surjective map. These conjectures can be proved in some cases: we survey the known results and add some new ones. The conjectures fail if they are generalized to nonrealizable oriented matroids ℳn.
Article PDF
Similar content being viewed by others
References
[AL] L. Allys and M. Las Vergnas: Minors of matroid morphisms,J. Combin. Theory Ser. B (to appear).
[Ba1] E. K. Babson: A combinatorial flag space, Ph.D. Thesis, MIT, 1993.
[Ba2] E. K. Babson: Personal communication.
[BKS] L. J. Billera, M. M. Kapranov, and B. Sturmfels: Cellular strings on polytopes,Proc. Amer. Math. Soc., to appear.
[Bj] A. Björner: Topological methods, in:Handbook of Combinatorics (R. Graham, M. Grötschel, and L. Lovász, eds.), North-Holland, Amsterdam, to appear.
[BLS+] A. Björner, M. Las Vergnas, B. Sturmfels, N. White, and G. M. Ziegler:Oriented Matroids, Encyclopedia of Mathematics, Cambridge University Press, Cambridge, 1993.
[BZ] T. H. Brylawski and G. M. Ziegler: Topological representation of dual pairs of oriented matroids,Discrete Comput. Geom., this issue, pp. 237–240.
[FL] J. Folkman, and J. Lawrence: Oriented matroids,J. Combin. Theory Ser. B 25 (1978), 199–236.
[GM] I. M. Gel'fand and R. D. MacPherson: A combinatorial formula for the Pontrjagin classes,Bull. Amer. Math. Soc. 26 (1992), 304–309.
[H] D. Husemoller:Fiber Bundles, McGraw-Hill, New York, 1966.
[K] J. P. S. Kung: Strong maps, in:The Theory of Matroids (N. White, ed.), Cambridge University Press, Cambridge, 1986, pp. 224–253.
[Ma] R. D. MacPherson: Combinatorial differential manifolds, Preprint, 1992.
[M] N. E. Mnëv: The universality theorems on the classification problem of configuration varieties and convex polytopes varieties, in:Topology and Geometry—Rohlin Seminar (O. Ya Viro, ed.), Lecture Notes in Mathematics, Vol. 1346, Springer-Verlag, Berlin, 1988, pp. 527–544.
[MR] N. E. Mnëv and J. Richter-Gebert: Two constructions of oriented matroids with disconnected extension space,Discrete Comput. Geom., this issue, pp. 271–285.
[R] J. Richter-Gebert: Oriented matroids with few mutations,Discrete Comput. Geom., this issue, pp. 251–269.
[S] M. Salvetti: Topology of the complement of real hyperplanes in ℂN.Invent. Math. 88 (1987), 603–618.
[SZ] B. Sturmfels and G. M. Ziegler: Extension spaces of oriented matroids,Discrete Comput. Geom. 10(1), (1993), 23–45.
Author information
Authors and Affiliations
Additional information
Nicolai E. Mnëv was partially supported by the Institut Mittag-Leffler.
Rights and permissions
About this article
Cite this article
Mnëv, N.E., Ziegler, G.M. Combinatorial models for the finite-dimensional Grassmannians. Discrete Comput Geom 10, 241–250 (1993). https://doi.org/10.1007/BF02573979
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02573979