Cones over the boundaries of nonshellable but constructible 3-balls
At present, the only known way (via short knots; see \textit{M. Hachimori} and \textit{G. M. Ziegler} [Math. Z. 235, No. 1, 159--171 (2000; Zbl 0965.57022)]) to construct nonshellable simplicial \(3\)-spheres yields examples that are nonconstructible. Thus, it remains unclear whether there are nonshellable but constructible \(3\)-spheres. In this paper, the author investigates \(3\)-spheres that are obtained as the cones over nonshellable but constructible \(3\)-balls. It is shown that if a (constructible) simplicial \(3\)-ball can be decomposed into two shellable \(3\)-balls such that their intersection is a \(2\)-ball, then the cone over its boundary yields a shellable \(3\)-sphere. Shellability of the resulting \(3\)-spheres is also proved for the case of more complicated constructible \(3\)-balls.
scientific article; zbMATH DE number 2111462