The Generalized Baues Problem for Cyclic Polytopes I

An important special case of the generalized Baues problem asks whether the order complex of all proper polyhedral subdivisions of a given point configuration, partially ordered by refinement, is homotopy equivalent to a sphere. In this paper, an affirmative answer is given for the vertex sets of cyclic polytopes in all dimensions. This yields the first non-trivial class of point configurations with neither a bound on the dimension, the codimension, nor the number of vertices for which this is known to be true. Moreover, it is shown that all triangulations of cyclic polytopes are lifting triangulations. This contrasts the fact that in general there are many non-regular triangulations of cyclic polytopes. Beyond this, we find triangulations of C(11, 5) with flip deficiency. This proves—among other things—that there are triangulations of cyclic polytopes that are non-regular for every choice of points on the moment curve.