The E t-Construction for Lattices, Spheres and Polytopes

We describe and analyze a new construction that produces new Eulerian lattices from old ones. It specializes to a construction that produces new strongly regular cellular spheres (whose face lattices are Eulerian). The construction does not always specialize to convex polytopes; however, in a number of cases where we can realize it, it produces interesting classes of polytopes. Thus we produce an infinite family of rational 2-simplicial 2-simple 4-polytopes, as requested by Eppstein, Kuperberg and Ziegler [6]. We also construct for each d ≥ 3 an infinite family of (d− 2)-simplicial 2-simple d-polytopes, thus solving a problem of Grünbaum [9].