2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/65845We construct 2^{Ω(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 >= 4.9 pages, 4 figures; Incorporated referee's comments; to appear in Math. AnnalenMetric GeometryCombinatoricsGeometric Topology52B11; 52B70, 57Q15text