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Discrete & Computational Geometry

, Volume 10, Issue 3, pp 241–250 | Cite as

Combinatorial models for the finite-dimensional Grassmannians

  • Nicolai E. Mnëv
  • Günter M. Ziegler
Article

Abstract

Let ℳ n be a linear hyperplane arrangement in ℝ n . We define two corresponding posetsG k (ℳ n andV k (ℳ n ) of oriented matroids, which approximate the GrassmannianG k (ℝ n ) and the Stiefel manifoldV k (ℝ n ). The basic conjectures are that the “OM-Grassmannian”G k (ℳ n ) has the homotopy type ofG k (ℝ n ), and that the “OM-Stiefel bundle” Δπ: ΔV k (ℳ n ) → ΔG k (ℳ n ) 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 ℳ n .

Keywords

Discrete Comput Geom Homotopy Type Maximal Chain Combinatorial Model Extension Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1993

Authors and Affiliations

  • Nicolai E. Mnëv
    • 1
  • Günter M. Ziegler
    • 2
  1. 1.St. PetersburgRussia
  2. 2.Institut Mittag-LefflerDjursholmSweden

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