Reisner's criterion give a complete characterization of Cohen–Macaulay simplicial complexes, based on $link$s of faces of the simplicial complex. Is there a known fact that relate shellability of a simplicial complex to the $link$s of its faces?
Thank you.
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If $\Delta$ is a shellable simplicial complex, then for any face $\sigma \in \Delta$ the link $\mathrm{ln}_{\Delta} \sigma$ is also a shellable simplicial complex. This is because we can the order the facets of $\mathrm{ln}_{\Delta} \sigma$ by taking ordering induced by the ordering of the facets of $\Delta$ which gives the shelling. See Proposition 10.14 of Shellable Nonpure Complexes and Posets II by Bjorner and Wachs available here.
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$\begingroup$ Since the link of the empty face is the complex itself, it is an if and only if. $\endgroup$ May 17, 2016 at 3:14