Face Numbers of 4-Polytopes and 3-Spheres

Steinitz (1906) gave a remarkably simple and explicit description of the set of all f -vectors f(P ) = (f0, f1, f2) of all 3-dimensional convex polytopes. His result also identifies the simple and the simplicial 3-dimensional polytopes as the only extreme cases. Moreover, it can be extended to strongly regular CW 2-spheres (topological objects), and further to Eulerian lattices of length 4 (combinatorial objects). The analogous problems “one dimension higher,” about the f -vectors and flag-vectors of 4-dimensional convex polytopes and their generalizations, are by far not solved, yet. However, the known facts already show that the answers will be much more complicated than for Steinitz’ problem. In this lecture, we will summarize the current state of knowledge. We will put forward two crucial parameters of fatness and complexity : Fatness F(P ) := f1+f2−20 f0+f3−10 is large if there are many more edges and 2-faces than there are vertices and facets, while complexity C(P ) := f03−20 f0+f3−10 is large if every facet has many vertices, and every vertex is in many facets. Recent results suggest that these parameters might allow one to differentiate between the cones of f or flag-vectors of • connected Eulerian lattices of length 5 (combinatorial objects), • strongly regular CW 3-spheres (topological objects), • convex 4-polytopes (discrete geometric objects), and • rational convex 4-polytopes (whose study involves arithmetic aspects). Further progress will depend on the derivation of tighter f -vector inequalities for convex 4-polytopes. On the other hand, we will need new construction methods that produce interesting polytopes which are far from being simplicial or simple — for example, very “fat” or “complex” 4-polytopes. In this direction, I will report about constructions (from joint work with Michael Joswig, David Eppstein and Greg Kuperberg) that yield • strongly regular CW 3-spheres of arbitrarily large fatness, • convex 4-polytopes of fatness larger than 5.048, and • rational convex 4-polytopes of fatness larger than 5− ε. 2000 Mathematics Subject Classification: 52B11, 52B10, 51M20