Face numbers of 4-Polytopes and 3-Spheres
| dc.creator | Ziegler, Günter M. | |
| dc.date | 2002-08-09 | |
| dc.date | 2003-06-20 | |
| dc.date.accessioned | 2026-07-07T04:50:09Z | |
| dc.date.available | 2026-07-07T04:50:09Z | |
| dc.description | 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. | |
| dc.identifier | https://arxiv.org/abs/math/0208073 | |
| dc.identifier | http://arxiv.org/abs/math/0208073 | |
| dc.identifier | Proceedings of the ICM, Beijing 2002, vol. 3, 625--636 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/64691 | |
| dc.subject | Metric Geometry | |
| dc.subject | Combinatorics | |
| dc.subject | 52B11, 52B10, 51M20 | |
| dc.title | Face numbers of 4-Polytopes and 3-Spheres | |
| dc.type | text |