Connectivity through bounds for the Castelnuovo-Mumford regularity

TL;DR

This paper introduces a method linking the connectivity of simplicial complexes' 1-skeleta to bounds on Castelnuovo-Mumford regularity, unifying previous results and providing tight bounds for pseudomanifolds.

Contribution

It generalizes and unifies existing connectivity results by relating them to Castelnuovo-Mumford regularity bounds for Stanley-Reisner rings.

Findings

Connectivity bounds for 1-skeleta of simplicial complexes.

Unified framework for previous connectivity results.

Proved the tightness of the connectivity bounds.

Abstract

We present a simple method to obtain information regarding the connectivity of the 1-skeleta of a wide family of simplicial complexes through bounds for the Castelnuovo-Mumford regularity of their Stanley-Reisner rings. In this way we generalize and unify two results on connectivity: one by Balinsky and Barnette, one by Athanasiadis. In particular, if $\Delta$ is a simplicial $d$-pseudomanifold, and $s$ is the highest integer such that there is an $s$-dimensional simplex not contained in $\Delta$, but such that its boundary is, then the 1-skeleton of $\Delta$ is $\left\lceil \frac{(s+1)d}{s} \right\rceil$-connected. We also show that this bound on the connectivity is tight.