Unshellable triangulations of spheres
To shell a triangulated \(n\)-disk or \(n\)-sphere means to remove its \(n\)-simplices one by one so that after each step (except the last) a triangulated \(n\)-disk remains. It is easy to see that for \(n\leq 2\) every triangulation of the \(n\)-disk or the \(n\)-sphere can be shelled, and it has been generally known that for \(n\geq 3\) unshellable triangulations exist. But whereas explicit constructions of unshellable triangulations of disks have been available in the literature [\textit{M.-E. Rudin}, Bull. Am. Math. Soc. 64, 90--91 (1958; Zbl 0082.37602); \textit{R. H. Bing}, Lect. on Modern Math. 2, 93--128 (1964; Zbl 0126.39104); \textit{R. E. Goodrick}, Proc. Camb. Philos. Soc. 64, 31--36 (1968; Zbl 0179.52602)], those of spheres have apparently not been. In the present article the author proves that for each \(n\geq 3\) there exists a combinatorial triangulation of the \(n\)-sphere such that after removing any \(n\)-simplex from it we are left with a noncollapsible (and a fortiori unshellable) triangulation of the \(n\)-disk.
noncollapsible triangulation of the \(n\)-disk
combinatorial triangulation of the \(n\)-sphere
scientific article; zbMATH DE number 29893
scientific article