Four-dimensional football, fullerenes and diagram geometry

An n-fullerene is an n-dimensional cell complex where the stars of the points are (n − 1)-dimensional simplices and the 2-cells are pentagons or hexagons. We say that an n-fullerene is uniform if the number of hexagonal faces containing a given vertex p (and, when n¿ 3, contained in a given 3-face X on p) does not depend on the choice of p (and X ). For instance, the dodecahedron, the truncated icosahedron (also called the football) and the tesselation of the euclidean plane in regular hexagons are uniform fullerenes. In this paper, we exploit notions and results of diagram geometry to classify /nite uniform fullerenes. In particular, we prove that there is no four-dimensional analogue of the football. More precisely, we prove that there is just one simply connected 4-fullerene where the cells are truncated icosahedra, but it is obtained as a Grassmann geometry of a non-spherical (whence, in/nite) Coxeter complex. Being in/nite, that fullerene is not a polytope. c © 2001 Elsevier Science B.V. All rights reserved.