[For the entire collection see Zbl 0699.00014.]
From the introduction: “Suppose that a finite simplicial complex \(C\) has \(f_ i\) faces of dimension \(i\). The sequence \((f_ 0,f_ 1,\dots)\) is called the \(f\)-vector of \(C\). Suppose now that \(\Gamma\) is a class of simplicial complexes. Is it possible to characterize the \(f\)-vectors of complexes in \(\Gamma\)?
This question has been often studied for various classes \(\Gamma\), and a number of characterization theorems are known. These are (in roughly chronological order; details are given in the paper): (1) for all complexes, (2) for Cohen-Macaulay complexes, (3) for boundary complexes of simplicial convex polytopes, (4) for Leray complexes, and (5) for complexes having a given sequence of Betti numbers. For some other classes, partial information and conjectures are available.
Most of the classes for which characterization efforts have been successful share the property that they can be defined in terms of homological conditions on all or some quotient complexes (links of faces). In this paper we review some of the advances in the study of \(f\)- vectors from this common point of view”.