Obstructions to shellability, partitionability, and sequential Cohen-Macaulayness

TL;DR

This paper characterizes all minimal obstructions to shellability, partitionability, and sequential Cohen-Macaulayness in low-dimensional simplicial complexes, revealing their equivalence in certain classes like flag complexes.

Contribution

It explicitly determines all obstructions to shellability in dimensions up to 2 and shows the equivalence of obstructions to three key properties in these classes.

Findings

Obstructions to shellability, partitionability, and sequential Cohen-Macaulayness coincide in dimensions ≤ 2.

These obstructions also coincide within the class of flag complexes.

Hereditary properties of shellability, partitionability, and sequential Cohen-Macaulayness are equivalent in these classes.

Abstract

For a property $\cal P$ of simplicial complexes, a simplicial complex $\Gamma$ is an obstruction to $\cal P$ if $\Gamma$ itself does not satisfy $\cal P$ but all of its proper restrictions satisfy $\cal P$. 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.