SC 98-14 Jörg Rambau, Francisco Santos: The Generalized Baues Problem for Cyclic Polytopes
Abstract: The Generalized Baues Problem asks whether for a given point
configuration the order complex of all its proper polyhedral subdivisions, 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 vertice 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.
Keywords: Generalized Baues Problem,
Polyhedral Subdivisions,
Induced Subdivisions,
Poset,
Spherical,
Cyclic Polytopes,
Bistellar Operations,
Flip Deficiency
MSC: 52C22, 52B99