2002/08/31 by Günter M. Ziegler · 1 citation
Mathematics · #math.MG #math.CO #msc:52B11 #msc:52B10 #msc:51M20
published as Proceedings of the ICM, Beijing 2002, vol. 3, 625--636
arxiv created 2003/06/20 · arxiv updated 2009/11/30
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) := (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-epsilon.