On the Shrinkable U.S.C. Decomposition Spaces of Spheres

Abstract

Let $G$ be a u.s.c decomposition of $S^n$, $H_G$ denote the set of nondegenerate elements and $\pi$ 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 $\pi (H_G)$, and such frontiers satisfies the Mismatch Property. Then this paper shows that this condition implies $S^n/G$ is homeomorphic to $S^n$ ($n\ge 4$). 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].

arXiv comment10 pages, 1 figure, Revised the proof of Lemma 2 and some typos. Readers can find a weaker form of the question discussed in the article by referring arXiv: 1309.5400. This paper will be presented in Special Session on Geometric Topology I, Southeastern Spring Sectional Meeting, Knoxville, Tennessee

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