  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.
