On stars and links of shellable polytopal complexes. (Q2490865)

If \(\mathcal C\) is a shellable simplicial complex and \(v\) is a vertex of \(\mathcal C\), then the star of \(v\) and the link of \(v\) in \(\mathcal C\) are both shellable simplicial complexes. Suppose \(\mathcal C\) is a polytopal complex, that is, a finite collection of convex polytopes in Euclidean space such that every face of a polytope in \(\mathcal C\) is in \(\mathcal C\), and the intersection of any two polytopes in \(\mathcal C\) is a face of each. If \(\mathcal C\) is shellable, are the stars and the links of vertices shellable? The question is posed in [\textit{G. M. Ziegler}, Lectures on polytopes (Graduate Texts in Mathematics. 152. Springer-Verlag, Berlin) (1995; Zbl 0823.52002)], and is addressed in this paper. The star of a vertex \(v\) in a polytopal complex \(\mathcal C\) is the set of all polytopes in \(\mathcal C\) containing \(v\) and all their faces. The author shows that the restriction of any shelling order of \(\mathcal C\) to the facets of the star of \(v\) in \(\mathcal C\) is a shelling order for the star. For links the answer is not so simple. There are two notions of link. The spherical link of a vertex \(v\) is a polytopal complex whose face poset is the union of all intervals \([v,H]\), where \(H\) is a polytope of \(\mathcal C\) containing \(v\). (If \(\mathcal C\) is the face complex of a polytope, this is the vertex figure of \(\mathcal C\).) The spherical link of a vertex in a shellable complex is shellable. However, this is different from the link of \(v\), which is the boundary of the star of \(v\). Here a strong condition on a shelling of the star implies shellability of the link. In particular, if all facets of the star are simple polytopes, then the link is shellable. The author suspects that shellable polytopal complexes of dimension 4 or higher can have nonshellable links.