Shrinkability of Decomposition of Sn Having Arbitrarily Small Neighborhoods with (n − 1)-Sphere Frontiers

Let G be a usc decomposition of Sn, HG denote the set of nonde-generate elements and pi be the natural projection of Sn onto Sn/G. Suppose that each point in the decomposition space has arbitrarily small neighborhoods with (n − 1)-sphere frontiers or boundaries which miss pi(HG). If all the arcs are tame in the particular area on the boundary of an n-cell C in Sn, then this paper shows that this con-dition implies Sn/G is homeomorphic to Sn (n ≥ 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].