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

The universality theorem for neighborly polytopes

  • Karim A. Adiprasito
  • Arnau Padrol
Original Paper


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 


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  1. [1]
    A. Altshuler and L. Steinberg: Neighborly 4-polytopes with 9 vertices, J. Combinatorial Theory Ser. A 15 (1973), 270–287.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetzbMATHGoogle Scholar
  4. [4]
    J. Bokowski and B. Sturmfels: On the coordinatization of oriented matroids, Discrete Comput. Geom. 1 (1986), 293–306.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    H. Günzel: The universal partition theorem for oriented matroids, Discrete Comput. Geom. 15 (1996), 121–145.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    N. E. Mnëv: On manifolds of combinatorial types of projective configurations and convex polyhedra, Sov. Math., Dokl. 32 (1985), 335–337.zbMATHGoogle 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, phd1.pdf.zbMATHGoogle 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.MathSciNetzbMATHGoogle Scholar
  11. [11]
    A. Padrol: Many neighborly polytopes and oriented matroids, Discrete Comput. Geom. 50 (2013), 865–902.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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

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