Connectivity through bounds for the Castelnuovo-Mumford regularity

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.