Obstructions to shellability, partitionability, and sequential Cohen-Macaulayness

Details Obstructions to shellability, partitionability, and sequential Cohen-Macaulayness Obstructions to shellability, partitionability, and sequential Cohen-Macaulayness

2010-03-10

arxiv.org/abs/1003.2037v2

Mathematics

Combinatorics

Scientific paper

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.

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