Face enumeration on simplicial complexes

TL;DR

This paper surveys recent advances in understanding the face numbers of triangulations of manifolds, exploring how topology and combinatorial conditions influence possible face configurations.

Contribution

It provides a comprehensive overview of new theorems and ideas in face enumeration of manifold triangulations, highlighting recent progress in the field.

Findings

Recent theorems restrict face numbers based on topology.

Combinatorial conditions like flagness impose additional constraints.

The field has experienced rapid growth in the last decade.

Abstract

Let $M$ be a closed triangulable manifold, and let $\Delta$ be a triangulation of $M$. What is the smallest number of vertices that $\Delta$ can have? How big or small can the number of edges of $\Delta$ be as a function of the number of vertices? More generally, what are the possible face numbers ($f$-numbers, for short) that $\Delta$ can have? In other words, what restrictions does the topology of $M$ place on the possible $f$-numbers of triangulations of $M$? To make things even more interesting, we can add some combinatorial conditions on the triangulations we are considering (e.g., flagness, balancedness, etc.) and ask what additional restrictions these combinatorial conditions impose. While only a few theorems in this area of combinatorics were known a couple of decades ago, in the last ten years or so, the field simply exploded with new results and ideas. Thus we feel that a survey paper is long overdue. As new theorems are being proved while we are typing this chapter, and as we have only a limited number of pages, we apologize in advance to our friends and colleagues, some of whose results will not get mentioned here.