17 Face Numbers of Polytopes and Complexes

Geometric objects are often put together from simple pieces according to certain combinatorial rules. As such, they can be described as complexes with their constituent cells, which are usually polytopes and often simplices. Many constraints of a combinatorial and topological nature govern the incidence structure of cell complexes and are therefore relevant in the analysis of geometric objects. Since these incidence structures are in most cases too complicated to be well understood, it is worthwhile to focus on simpler invariants that still say something nontrivial about their combinatorial structure. The invariants to be discussed in this chapter are the f -vectors f = (f0, f1, . . .), where fi is the number of i-dimensional cells in the complex. The theory of f -vectors can be discussed at two levels: (1) the numerical relations satisfied by the fi numbers, and (2) the algebraic, combinatorial, and topological facts and constructions that give rise to and explain these relations. This chapter will summarize the main facts in the numerology of f -vectors (i.e., at level 1), with emphasis on cases of geometric interest. The chapter is organized as follows. To begin with, we treat simplicial complexes, first the general case (Section 17.1), then complexes with various Betti number constraints (Section 17.2), and finally triangulations of spheres, polytope boundaries, and manifolds (Section 17.3). Then we move on to nonsimplicial complexes, discussing first the general case (Section 17.4) and then polytopes and spheres (Section 17.5).