Face numbers of 4-Polytopes and 3-Spheres

In this paper, we discuss f- and flag-vectors of 4-dimensional convex polytopes and cellular 3-spheres. We put forward two crucial parameters of fatness and complexity: Fatness F(P) := (f_1+f_2-20)/(f_0+f_3-10) is large if there are many more edges and 2-faces than there are vertices and facets, while complexity C(P) := (f_{03}-20)/(f_0+f_3-10) 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-epsilon.