N ov 2 00 3 Many Triangulated 3 - Spheres Julian
نویسندگان
چکیده
We construct 2Ω(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 2O(n logn) 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.
منابع مشابه
ar X iv : m at h . C O / 0 11 02 40 v 1 2 2 O ct 2 00 1 Kalai ’ s squeezed 3 - spheres are polytopal
In 1988, Kalai [5] extended a construction of Billera and Lee to produce many triangulated (d−1)-spheres. In fact, in view of upper bounds on the number of simplicial d-polytopes by Goodman and Pollack [2, 3], he derived that for every dimension d ≥ 5, most of these (d− 1)-spheres are not polytopal. However, for d = 4, this reasoning fails. We can now show that, as already conjectured by Kalai,...
متن کاملO ct 2 00 1 Kalai ’ s squeezed 3 - spheres are polytopal
In 1988, Kalai [5] extended a construction of Billera and Lee to produce many triangulated (d−1)-spheres. In fact, in view of upper bounds on the number of simplicial d-polytopes by Goodman and Pollack [2, 3], he derived that for every dimension d ≥ 5, most of these (d− 1)-spheres are not polytopal. However, for d = 4, this reasoning fails. We can now show that, as already conjectured by Kalai,...
متن کاملMany Triangulated 3-Spheres
We construct 2Ω(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 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...
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