Discrete & Computational Geometry

, Volume 10, Issue 1, pp 23–45 | Cite as

Extension spaces of oriented matroids

  • Bernd Sturmfels
  • Günter M. Ziegler


We study the space of all extensions of a real hyperplane arrangement by a new pseudohyperplane, and, more generally, of an oriented matroid by a new element. The question whether this space has the homotopy type of a sphere is a special case of the “Generalized Baues Problem” of Billera, Kapranov, and Sturmfels, via the Bohne-Dress theorem on zonotopal tilings.

We prove that the extension space is spherical for the class of strongly euclidean oriented matroids. This class includes the alternating matroids and all oriented matroids of rank at most 3 or of corank at most 2. In general it is not known whether the extension space is connected for all realizable oriented matroids (hyperplane arrangements). We show that the subspace of realizable extensions is always connected but not necessarily spherical. Nonrealizable oriented matroids of rank 4 with disconnected extension spaces were recently constructed by Mnëv and Richter-Gebert.


Homotopy Type Order Ideal Strong Component Hyperplane Arrangement 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.


  1. 1.
    E. K. Babson: A combinatorial flag space, Ph.D. Thesis, MIT, 1993.Google Scholar
  2. 2.
    A. Bachem and W. Kern: Adjoints of oriented matroids,Combinatorica 6 (1986), 299–308.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    L. J. Billera, M. M. Kapranov, and B. Sturmfels: Cellular strings on polytopes, preprint, 1991.Google Scholar
  4. 4.
    L. J. Billera and B. Sturmfels: Fiber polytopes,Ann. of Math. 135 (1992), 527–549.MathSciNetCrossRefGoogle Scholar
  5. 5.
    A. Björner: Topological methods, in:Handbook of Combinatorics (R. Graham, M. Grötschel, and L. Lovász, eds.), North-Holland, Amsterdam, to appear.Google Scholar
  6. 6.
    A. Björner, M. Las Vergnas, B. Sturmfels, N. White, and G. M. Ziegler:Oriented Matroids, Cambridge University Press, Cambridge, 1993.Google Scholar
  7. 7.
    J. Bohne and A. Dress: Penrose tilings and oriented matroids, in preparation.Google Scholar
  8. 8.
    J. Bokowski and J. Richter-Gebert: On the classification of non-realizable oriented matroids. Part I: Generation, Part II: Properties, preprints, 1990.Google Scholar
  9. 9.
    J. Edmonds and K. Fukuda: Oriented matroid programming, Ph.D. Thesis of K. Fukuda, University of Waterloo, 1982.Google Scholar
  10. 10.
    J. Edmonds and A. Mandel: Topology of oriented matroids, Ph.D. Thesis of A. Mandel, University of Waterloo, 1982.Google Scholar
  11. 11.
    J. Folkman and J. Lawrence: Oriented matroids,J. Combin. Theory Ser. B 25 (1978) 199–236.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    K. Fukuda and A. Tamura: Local deformation and orientation transformation in oriented matroids, I, II, Ars Combin.25A, 243–258; and preprint (Research Reports on Information Sciences B-212, Tokyo Institute of Technology), 1988.Google Scholar
  13. 13.
    I. M. Gelfand and R. D. MacPherson: A combinatorial formula for the Pontrjagin classes,Bull. Amer. Math. Soc. 26 (1992), 304–309.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    B. Grünbaum:Convex Polytopes, Interscience, London, 1967.zbMATHGoogle Scholar
  15. 15.
    M. M. Kapranov and V. A. Voevodsky: Combinatorial-geometric aspects of polycategory theory: pasting schemes and higher Bruhat orders (list of results),Cahiers Topologie Géom. Différentielle 32 (1991), 11–28.MathSciNetzbMATHGoogle Scholar
  16. 16.
    M. Las Vergnas: Extensions ponctuelles d'une géométrie combinatoire orientée, in:Problémes combinatoires et théorie des graphes (Actes Coll., Orsay, 1976), Colloques internationaux, C.N.R.S., No. 260 (1978), pp. 265–270.Google Scholar
  17. 17.
    F. Levi: Die Teilung der projektiven Ebene durch Geraden oder Pseudogeraden,Ber. Math.-Phys. Kl. Sächs. Akad. Wiss.,78 (1926), 256–267.Google Scholar
  18. 18.
    Y. I. Manin and V. V. Schechtman: Arrangements of hyperplanes, higher braid groups and higher Bruhat orders,Adv. Studies in Pure Math. 17 (1989), 289–308.MathSciNetGoogle Scholar
  19. 19.
    N. E. Mnëv and J. Richter-Gebert: Two Constructions of Oriented Matroids with Disconnected Extension Space,Discrete Comput. Geom., to appear.Google Scholar
  20. 20.
    N. E. Mnëv and G. M. Ziegler: Combinatorial Models for the Finite-Dimensional Grassmannians,Discrete Comput. Geom., to appear.Google Scholar
  21. 21.
    D. Quillen: Higher Algebraic K-Theory: I, in:Higher K-Theories (H. Bass, ed.), Lecture Notes in Mathematics, Vol. 341, Springer-Verlag, Berlin, 1973, pp. 85–147.CrossRefGoogle Scholar
  22. 22.
    J. Richter-Gebert: Euclideanness and Final Polynomials in Oriented Matroid Theory, Report No. 12 (1991/92), Institut Mittag-Leffler, 1991;Combinatorica, to appear.Google Scholar
  23. 23.
    P. Y. Suvorov: Isotopic but not rigidly isotopic plane systems of straight lines, in:Topology and Geometry-Rohlin Seminar (O. Ya Viro, ed), Lecture Notes in Mathematics, Vol. 1346, Springer-Verlag, Heidelberg, 1988, pp. 545–556.CrossRefGoogle Scholar
  24. 24.
    G. M. Ziegler: Higher Bruhat orders and cyclic hyperplane arrangements,Topology, in press.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1993

Authors and Affiliations

  • Bernd Sturmfels
    • 1
  • Günter M. Ziegler
    • 2
  1. 1.Department of MathematicsCornell UniversityIthacaUSA
  2. 2.Konrad-Zuse-Zentrum (ZIB)Berlin 31Federal Republic of Germany

Personalised recommendations