Abstract: It is known that the (2k?1)(2k?1) -sphere has at most 2O(nklogn)2O(nklog?n) combinatorially distinct triangulations with n vertices, for every k?2k?2 . Here we construct at least 2?(nk)2?(nk) such triangulations, improving on the previous constructions which gave 2?(nk?1)2?(nk?1) in the general case (Kalai) and 2?(n5/4)2?(n5/4) for k=2k=2 (Pfeifle?Ziegler). We also construct 2?(nk?1+1k)2?(nk?1+1k) geodesic (a.k.a. star-convex) n-vertex triangulations of the (2k?1)(2k?1) -sphere. As a step for this (in the case k=2k=2 ) we construct n-vertex 4-polytopes containing ?(n3/2)?(n3/2) facets that are not simplices, or with ?(n3/2)?(n3/2) edges of degree three.

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