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


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 .


Discrete Comput Geom Homotopy Type Maximal Chain Combinatorial Model Extension Space 
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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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