Obstructions to shellability, partitionability, and sequential Cohen-Macaulayness

Abstract

For a property $\mathcal{P}$ of simplicial complexes, a simplicial complex $\Gamma$ is an obstruction to $\mathcal{P}$ if $\Gamma$ itself does not satisfy $\mathcal{P}$ but all of its proper restrictions satisfy $\mathcal{P}$. In this paper, we determine all obstructions to shellability of dimension $\le 2$, refining the previous work by Wachs. As a consequence we obtain that the set of obstructions to shellability, that to partitionability and that to sequential Cohen-Macaulayness all coincide for dimensions $\le 2$. We also show that these three sets of obstructions coincide in the class of flag complexes. These results show that the three properties, hereditary-shellability, hereditary-partitionability, and hereditary-sequential Cohen-Macaulayness are equivalent for these classes.