Eran Nevo and Stedman Wilson have constructed triangulations with n vertices of the 3-dimensional sphere! This settled an old problem which stood open for several decades. Here is a link to their paper How many n-vertex triangulations does the 3 -sphere have?
1) Since the number of facets in an n-vertex triangulation of a 3-sphere is at most quadratic in n, an upper bound for the number of triangulations of the 3-sphere with n vertices is . For certain classes of triangulations, Dey removed in 1992 the logarithmic factor in the exponent for the upper bound.
2) Goodman and Pollack showed in 1986 that the number of simplicial 4-polytopes with n vertices is much much smaller . This upper bound applies to simplicial polytopes of every dimension d, and Alon extended it to general polytopes.
3) Before the new paper the world record was the 2004 lower bound by Pfeifle and Ziegler –
4) In 1988 I constructed triangulations of the d-spheres with n vertices. The new construction gives hope to improve it in any odd dimension by replacing [d/2] by [(d+1)/2] (which match up to logn the exponent in the upper bound). [Update (Dec 19) : this has now been achieved by Paco Santos (based on a different construction) and Nevo and Wilson (based on extensions of their 3-D constructions). More detailed to come.]
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