A non-shellable 3-sphere
Shellability of simplicial complexes has been a useful concept in polyhedral theory, in piecewise linear topology and more recently in connection with Cohen Macaulay rings. This paper initiates a study of shellability in a graph theoretic context. For certain graphs, called \(n\)-graphs, necessary and sufficient conditions for shellability are given. A correspondence is established between \(n\)-graphs and (\(n-1\))-dimensional simplicial complexes. Via this correspondence, results on shellability of \(n\)-graphs have implications for problems in PL-topology. For example, it is known that all 2-spheres and convex \(d\)-spheres are shellable. It has been conjectured that all 3-spheres are shellable. A counter-example is provided in the following sense. A finite cell complex is constructed whose cells are simplices and such that the underlying space is a 3-sphere whose simplices cannot be shelled.
scientific article; zbMATH DE number 3922681
scientific article