There are two related poset structures, the higher Stasheff-Tamari orders, on the set of all triangulations of the cyclic d polytope with n vertices. In this paper it is shown that both of them have the homotopy type of a sphere of dimension n - d - 3. Moreover, we resolve positively a new special case of the Generalized Baues Problem: the Baues poset of all polytopal decompositions of a cyclic polytope of dimension d ≤ 3 has the homotopy type of a sphere of dimension n - d - 2.
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†The author was supported at MSRI in part by NSF grant #DMS 9022140. ‡The author was supported by a University of Minnesota McKnight-Land Grant Fellowship.