Combinatorica

, Volume 37, Issue 2, pp 129–136 | Cite as

The universality theorem for neighborly polytopes

Original Paper
  • 62 Downloads

Abstract

In this note, we prove that every open primary basic semialgebraic set is stably equivalent to the realization space of a neighborly simplicial polytope. This in particular provides the final step for Mnëv‘s proof of the universality theorem for simplicial polytopes.

Mathematics Subject Classification (2000)

52B40 52C40 14P10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A. Altshuler and L. Steinberg: Neighborly 4-polytopes with 9 vertices, J. Combinatorial Theory Ser. A 15 (1973), 270–287.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    A. Björner, M. Las Vergnas, B. Sturmfels, N. White, and G. M. Ziegler: Oriented matroids, second ed., Encyclopedia of Mathematics and its Applications, vol. 46, Cambridge University Press, Cambridge, 1999.Google Scholar
  3. [3]
    J. Bokowski and A. Guedes de Oliveira: Simplicial convex 4-polytopes do not have the isotopy property, Portugal. Math. 47 (1990), 309–318.MathSciNetMATHGoogle Scholar
  4. [4]
    J. Bokowski and B. Sturmfels: On the coordinatization of oriented matroids, Discrete Comput. Geom. 1 (1986), 293–306.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    H. Günzel: The universal partition theorem for oriented matroids, Discrete Comput. Geom. 15 (1996), 121–145.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    B. Jaggi, P. Mani-Levitska, B. Sturmfels and N. White: Uniform oriented matroids without the isotopy property, Discrete Comput. Geom. 4 (1989), 97–100.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    U. H. Kortenkamp: Every simplicial polytope with at most d+4 vertices is a quotient of a neighborly polytope, Discrete Comput. Geom. 18 (1997), 455–462.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    N. E. Mnëv: On manifolds of combinatorial types of projective configurations and convex polyhedra, Sov. Math., Dokl. 32 (1985), 335–337.MATHGoogle Scholar
  9. [9]
    N. E. Mnëv: The topology of configuration varieties and convex polytopes varieties, Ph.D. thesis, St. Petersburg State University, St. Petersburg, RU, 1986, 116 pages, pdmi.ras.ru/~mnev/mnev phd1.pdf.MATHGoogle Scholar
  10. [10]
    N. E. Mnëv: The universality theorems on the classification problem of configuration varieties and convex polytopes varieties, Topology and geometry, Rohlin Semin. 1984-1986, Lect. Notes Math. 1346, 527–543, 1988.MathSciNetMATHGoogle Scholar
  11. [11]
    A. Padrol: Many neighborly polytopes and oriented matroids, Discrete Comput. Geom. 50 (2013), 865–902.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    J. Richter-Gebert: Mnëv’s universality theorem revisited, Sém. Lothar. Combin. 34 (1995), Art. B34h, (electronic).Google Scholar
  13. [13]
    J. Richter-Gebert: Realization Spaces of Polytopes, Lecture Notes in Mathematics, vol. 1643, Springer, Berlin, 1996.Google Scholar
  14. [14]
    J. Richter-Gebert and G. M. Ziegler: Realization spaces of 4-polytopes are universal, Bulletin of the American Mathematical Society 32 (1995), 403–412.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    J. Richter-Gebert: The universality theorems for oriented matroids and poly-topes, Advances in discrete and computational geometry (South Hadley, MA, 1996), Contemp. Math., vol. 223, Amer. Math. Soc., Providence, RI, 1999, 269–292.Google Scholar
  16. [16]
    I. Shemer: Neighborly polytopes, Israel J. Math. 43 (1982), 291–314.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    P. W. Shor: Stretchability of pseudolines is NP-hard, Applied Geometry and Discrete Mathematics — The Victor Klee Festschrift (P. Gritzmann and B. Sturmfels, eds.), DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 4, Amer. Math. Soc., Providence RI, 1991, 531–554.Google Scholar
  18. [18]
    B. Sturmfels: Neighborly polytopes and oriented matroids, European J. Combin. 9 (1988), 537–546.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    B. Sturmfels: Simplicial polytopes without the isotopy property, preprints of the Institute for Mathematics and Applications (1988), 5.Google Scholar
  20. [20]
    G. M. Ziegler: Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152, Springer, New York, 1995, Revised edition, 1998; seventh updated printing 2007.Google Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Einstein Institute for MathematicsHebrew University of JerusalemJerusalemIsrael
  2. 2.Institut de Mathématiques de Jussieu - Paris Rive GaucheUniversit Pierre et Marie Curie (Paris 06)ParisFrance

Personalised recommendations