Many triangulated 3-spheres

Abstract.

We construct combinatorial types of triangulated 3-spheres on n vertices. Since by a result of Goodman and Pollack (1986) there are no more than 2O(n log n) combinatorial types of simplicial 4-polytopes, this proves that asymptotically there are far more combinatorial types of triangulated 3-spheres than of simplicial 4-polytopes on n vertices. This complements results of Kalai (1988), who had proved a similar statement about d-spheres and (d+1)-polytopes for fixed d≥4.

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References

  1. 1.

    Barnette, D.: Diagrams and Schlegel diagrams, in Combinatorial Structures and Their Applications. Proc. Calgary Internat. Conference 1969, New York, 1970, Gordon and Breach, pp. 1–4

  2. 2.

    Bender, E.A.: The number of three-dimensional convex polyhedra. Am. Math. Monthly 94, 7–21 (1987)

    MATH  Google Scholar 

  3. 3.

    Biggs, N.: Automorphisms of imbedded graphs. J. Comb. Theory Ser. B, 11, 132–138 (1971)

    Google Scholar 

  4. 4.

    Brahana, T.: Systems of circuits on 2-dimensional manifolds. Ann. Math. 23, 144–168 (1921)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Brückner, J.M.: Über die Ableitung der allgemeinen Polytope und die nach Isomorphismus verschiedenen Typen der allgemeinen Achtzelle (Oktatope). Verhand. Konik. Akad. Wetenschap, Erste Sectie, 10, 1910

  6. 6.

    Durhuus, B., Jonsson, T.: Remarks on the entropy of 3-manifolds. Nuclear Physics, Ser. B, 445, 182–192 (1995)

    Google Scholar 

  7. 7.

    Eppstein, D., Kuperberg, G., Ziegler, G.M.: Fat 4-polytopes and fatter 3-spheres, in Discrete Geometry: In honor of W. Kuperberg’s 60th birthday, A. Bezdek, (ed.), Pure and Applied Mathematics. A series of Monographs and Textbooks, Marcel Dekker Inc. 2003, pp. 239–265

  8. 8.

    Erdős, P.: Über die Primzahlen gewisser arithmetischer Reihen (German). Math. Z. 1935, pp. 473–491

  9. 9.

    Ewald, G.: Combinatorial Convexity and Algebraic Geometry. vol. 168 of Graduate Texts in Mathematics. Springer, 1996

  10. 10.

    Goodman, J.E., Pollack, R.: There are asymptotically far fewer polytopes than we thought. Bull. Am. Math. Soc. 14, 127–129 (1986)

    MATH  Google Scholar 

  11. 11.

    Goodman, J.E., Pollack, R.: Upper bounds for configurations and polytopes in ℝd. Discrete Comput. Geom. 1, 219–227 (1986)

    Google Scholar 

  12. 12.

    Gromov, M.: Spaces and questions, in GAFA 2000, Special Volume, Part I. N. Alon, J. Bourgain, A. Connes, M. Gromov, V. Milman, (eds.), Birkhäuser, 2000, pp. 118–161

  13. 13.

    Grünbaum, B., Sreedharan, V.: An enumeration of simplicial 4-polytopes with 8 vertices. J. Comb. Theory, 2, 437–465 (1967)

    Google Scholar 

  14. 14.

    Heffter, L.: Ueber metacyklische Gruppen und Nachbarconfigurationen. (German). Math. Ann. 50, 261–268 (1898)

    MATH  Google Scholar 

  15. 15.

    James, L.D., Jones, G.A.: Regular orientable imbeddings of complete graphs. J. Comb. Theory Ser. B, 39, 353–367 (1985)

    Google Scholar 

  16. 16.

    Kalai, G.: Many triangulated spheres. Discrete Comput. Geom. 3, 1–14 (1988)

    Google Scholar 

  17. 17.

    Lazarus, F., Pocchiola, M., Vegter, G., Verroust, A.: Computing a canonical polygonal schema of an orientable triangulated surface. In: Proc. 17th Ann. ACM Sympos. Comput. Geom. 2001, pp. 80–89

  18. 18.

    Pfeifle, J.: Kalai’s squeezed 3-spheres are polytopal. Discrete Comput. Geom. 27, 395–407 (2002)

    MATH  Google Scholar 

  19. 19.

    Stanley, R.P.: The upper-bound conjecture and Cohen-Macaulay rings. Stud. Appl. Math. 54, 135–142 (1975)

    MATH  Google Scholar 

  20. 20.

    Steinitz, E.: Polyeder und Raumeinteilungen. In: Encyclopädie der mathematischen Wissenschaften, Band 3 (Geometrie), Teil 3AB12, 1922, pp. 1–139

  21. 21.

    Steinitz, E., Rademacher, H.: Vorlesungen über die Theorie der Polyeder unter Einschluß der Elemente der Topologie. vol. 41 of Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin, 1934

  22. 22.

    Stillwell, J.: Classical Topology and Combinatorial Group Theory, no. 72 in Graduate Texts in Mathematics. Springer, second ed. 1993

  23. 23.

    Tutte, W.T.: On the enumeration of convex polyhedra. J. Comb. Theory Ser. B, 28, pp. 105–126 (1980)

    Google Scholar 

  24. 24.

    Wilson, S.: Families of regular maps in graphs. J. Comb. Theory Ser. B, 85, 269–289 (2002)

    Google Scholar 

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Correspondence to Günter M. Ziegler.

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Mathematics Subject Classification (1991): 52B11, 52B70, 57Q15

Supported by the Deutsche Forschungsgemeinschaft within the European graduate program Combinatorics, Geometry, and Computation (GRK 588/1) and an MSRI post-doctoral fellowship.

Partially supported by Deutsche Forschungs-Gemeinschaft (DFG), via the DFG Research Center “Mathematics in the Key Technologies” (FZT86), the Research Group “Algorithms, Structure, Randomness” (Project ZI 475/3), and a Leibniz grant (ZI 475/4).

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Pfeifle, J., Ziegler, G. Many triangulated 3-spheres. Math. Ann. 330, 829–837 (2004). https://doi.org/10.1007/s00208-004-0594-2

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Keywords

  • Similar Statement
  • Combinatorial Type