Shrinkability of Decomposition of $S^n$ Having Arbitrarily Small Neighborhoods with ($n-1$)-Sphere Frontiers

Abstract

Let $G$ be a usc decomposition of $S^n$, $H_G$ denote the set of nondegenerate elements and $π$ be the natural projection of $S^n$ onto $S^n/G$. Suppose that each point in the decomposition space has arbitrarily small neighborhoods with ($n-1$)-sphere frontiers or boundaries which miss $π(H_G)$. If all the arcs are tame in the particular area on the boundary of an $n$-cell $C$ in $S^n$, then this paper shows that this condition implies $S^n/G$ is homeomorphic to $S^n$ ($n\geq 4$). This answers a weak form of a conjecture asked by Daverman [2, p. 61]. In the case of $n=3$, the strong form of the conjecture has an affirmative answer from Woodruff [11].