Many triangulated 3-spheres


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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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).

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