On Stanley-Reisner Rings with Minimal Betti Numbers

TL;DR

This paper investigates simplicial complexes with minimal Betti numbers in their Stanley-Reisner rings, revealing unique combinatorial and topological properties that enable classification and inductive construction.

Contribution

It characterizes simplicial complexes with minimal Betti numbers, showing their Betti numbers follow binomial coefficients and their subcomplexes are homotopy equivalent to points or spheres.

Findings

Betti numbers are given by binomial coefficients

Full subcomplexes are homotopy equivalent to a point or a sphere

Allows classification and inductive construction of these complexes

Abstract

We study simplicial complexes with a given number of vertices whose Stanley-Reisner ring has the minimal possible Betti numbers. We find that these simplicial complexes have very special combinatorial and topological structures. For example, the Betti numbers of their Stanley-Reisner rings are given by the binomial coefficients, and their full subcomplexes are homotopy equivalent either to a point or to a sphere. These properties make it possible for us to either classify them or construct them inductively from instances with fewer vertices.