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arXiv · 1412.5920

Connectivity through bounds for the Castelnuovo-Mumford regularity

Gabriele Balletti
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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.

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Gabriele Balletti. 2014-12-18. Connectivity through bounds for the Castelnuovo-Mumford regularity. https://doi.org/10.1016/j.jcta.2016.11.011

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