This paper studies higher dimensional analogues of the Tamari lattice on triangulations of a convex n-gon, by placing a partial order on the triangulations of a cyclic d-polytope. Our principal results are that in dimension d ≤ 3, these posets are lattices whose intervals have the homotopy type of a sphere or ball, and in dimension d ≤ 5, all triangulations of a cyclic d-polytope are connected by bistellar operations.
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Acknowledgements. The authors thank Lou Billera, Jesus de Loera, Herbert Edelsbrunner, Bernd Sturmfels, and the referee for helpful comments and suggestions. In particular, we thank Jesus de Loera for his program PUNTOS, which computes the coherent triangulations of a point configuration si. We thank the U.S. NSF for some support for the first author by a grant and for a Mathematical Sciences Postdoctoral Research Fellowship for the second author.