Kalai's squeezed three-spheres are polytopal

Abstract

In 1988, Kalai [5] extended a construction of Billera and Lee to produce many triangulated(d−1)-spheres. In fact, in view of upper bounds on the number of simplicial d-polytopes by Goodman and Pollack [2,3], he derived that for every dimension d ≥ 5, most of these(d−1)-spheres are not polytopal. However, for d=4, this reasoning fails. We can now show that, as already conjectured by Kalai, all of his 3-spheres are in fact polytopal. We also give a shorter proof for Hebble and Lee’s result [4] that the dual graphs of these 4-polytopes are Hamiltonian.