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

Let G be a u.s.c decomposition of S n , H G denote the set of nondegenerate elements and π be the 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 which miss π ( H G ) , and such frontiers satisfy the Mismatch Property. Then this paper shows that this condition implies S n / G is homeomorphic to S n ( n ≥ 4 ). This answers a weakened form of a conjecture asked by Daverman [3, p. 61]. In the case n = 3 , the strong form of the conjecture has an affirmative answer from Woodruff [12].