Many triangulated 3-spheres

We construct 2(Omega(n5/4)) 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 dgreater than or equal to4.