Shrinking countable decompositions of $S^{3}$

Abstract:

Conditions are given which imply that a countable, cellular use decomposition $G$ is shrinkable. If the embedding of each element in $G$ has the bounded nesting property, defined in this paper, then ${S^3}/G$ is homeomorphic to ${S^3}$. The bounded nesting property is a condition on the defining sequence of cells for an element of $G$ which implies that $G$ satisfies the Disjoint Disk criterion for shrinkability [${\mathbf {S1}}$, Theorem 3.1]. From this result, one deduces that countable, star-like equivalent use decompositions of ${S^3}$ are shrinkable—a result proved independently by E. Woodruff [${\mathbf {W}}$]. Also, one deduces the shrinkability of countable bird-like equivalent use decompositions (a generalization of the star-like result), and the recently proved theorem that if each element of a countable use decomposition $G$ of ${S^3}$ has a mapping cylinder neighborhood, then $G$ is shrinkable [${\mathbf {E}}$; ${\mathbf {S1}}$, Theorem 4.1; ${\mathbf {S}}\text {-}{\mathbf {W}}$, Theorem 1].