Links and nonshellable cell partitionings of $S^ 3$

Abstract:

A cell partitioning of a closed $3$-manifold ${M^3}$ is a finite covering of ${M^3}$ by $3$cells that fit together in a bricklike pattern. A cell partitioning $H$ of ${M^3}$ is shellable if $H$ has a counting $\langle {h_1},{h_2}, \ldots ,{h_n}\rangle$ such that if $1 \leqslant i < n,\;{h_1} \cup {h_2} \cup \cdots \cup {h_i}$ is a $3$-cell. The main result of this paper is a relationship between nonshellability of a cell partitioning $H$ of ${S^3}$ and the existence of links in ${S^3}$ specially related to $H$. This result is used to construct a nonshellable cell partitioning of ${S^3}$.