On Stanley-Reisner Rings with Minimal Betti Numbers

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.

Figures (7)

• Figure 1: $\widetilde{D}$-minimal $1$-dimensional simplicial complexes with $5$ vertices
• Figure 2: $K$ is weakly tight but $\mathrm{link}_K u$ is not ($u\notin V_{\mathrm{mdim}}(K)$).
• Figure 3: All proper full subcomplexes of $K$ are weakly tight, but $K$ is not.
• Figure 4: A schematic picture of a weakly tight simplicial complex $K$ at a vertex $v\in V_{\mathrm{mdim}}(K)$
• Figure 5: Two different essential germ filtrations of $K$

Theorems & Definitions (74)

• Definition 1.1
• Remark 1.2
• Definition 1.3: Tight Simplicial Complex
• Theorem 1.4: see Theorem \ref{['Thm:Binom-Lower-Bound']}
• Remark 1.5
• Definition 1.6: Weakly Tight Simplicial Complex
• Corollary 1.7
• Theorem 1.8: see Theorem \ref{['Thm:Main-1-Repeat']}
• Corollary 1.9
• Theorem 1.10: see Theorem \ref{['Thm:Subcomplex-Tight']}