Günter Ziegler - Technische Universität Berlin

f-Vectors of 3-Spheres and 4-Polytopes

Abstract: Steinitz (1906) gave a complete characterization of the f-vectors (f_0,f_1,f_2) of the 3-dimensional convex polytopes: They are the integer points in a 2-dimensional convex polyhedral cone. His result is remarkably simple; it is easily extended to the larger generality of strongly regular cell decompositions of the 2-dimensional sphere, and of Eulerian lattices of length 4.

The analogous problems "in one dimension higher", about the f-vectors (f_0,f_1,f_2,f_3) and the flag vectors (f_0,f_1,f_2,f_3;f_{03}) of 4-dimensional convex polytopes, and of 3-dimensional regular CW spheres, are by far not solved, yet. They are in essence problems of 3-dimensional geometry! However, the known facts and available data already show that the answers will be much more complicated than for Steinitz' problem. In this lecture, we will summarize the current state of affairs. We highlight the crucial parameters of fatness and complexity. Recent results suggest that these parameters may allow one to differentiate between the f-vectors of

• Eulerian lattices of length 5 (combinatorial objects),
• strongly regular CW 3-spheres (topological objects),
• convex 4-polytopes (discrete geometric objects), as well as
• rational convex 4-polytopes (whose study involves arithmetic aspects).

• 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-\varepsilon.