On the shrinkable U.S.C. decomposition spaces of spheres

Let G   be a u.s.c decomposition of SnSn, HGHG denote the set of nondegenerate elements and π   be the projection of SnSn onto Sn/GSn/G. Suppose that each point in the decomposition space has arbitrarily small neighborhoods with (n−1n−1)-sphere frontiers which miss π(HG)π(HG), and such frontiers satisfy the Mismatch Property. Then this paper shows that this condition implies Sn/GSn/G is homeomorphic to SnSn (n≥4n≥4). This answers a weakened form of a conjecture asked by Daverman [3, p. 61]. In the case n=3n=3, the strong form of the conjecture has an affirmative answer from Woodruff [12].