Many Triangulated 3-Spheres

Details Many Triangulated 3-Spheres Many Triangulated 3-Spheres

2002-11-30

arxiv.org/abs/math/0212004v2

Mathematics

Metric Geometry

9 pages, 4 figures; Incorporated referee's comments; to appear in Math. Annalen

Scientific paper

We construct 2^{\Omega(n^{5/4})} combinatorial types of triangulated 3-spheres on n vertices. Since by a result of Goodman and Pollack (1986) there are no more than 2^{O(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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