## ABSTRACT

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 = ( f 0 , f 1 , ⋯ ) \$ f=(f_0, f_1, \dots ) \$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119601/fb8178cb-c53c-4311-b072-eff5ec016aba/content/inline-math17_1.tif"/> , where f i \$ f_i \$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119601/fb8178cb-c53c-4311-b072-eff5ec016aba/content/inline-math17_2.tif"/> is the number of i-dimensional cells in the complex.