Decomposition spaces having arbitrarily small neighborhoods with 2-sphere boundaries

Woodruff, Edythe P.

doi:10.1090/S0002-9947-1977-0442944-2

Decomposition spaces having arbitrarily small neighborhoods with $2$-sphere boundaries
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Trans. Amer. Math. Soc. 232 (1977), 195-204 Request permission

Abstract:

Let G be an u.s.c. decomposition of ${S^3}$. Let H denote the set of nondegenerate elements and P be the natural projection of ${S^3}$ onto ${S^3}/G$. Suppose that each point in the decomposition space has arbitrarily small neighborhoods with 2-sphere boundaries which miss $P(H)$. We prove in this paper that this condition implies that ${S^3}/G$ is homeomorphic to ${S^3}$. This answers a question asked by Armentrout [1, p. 15]. Actually, the hypothesis concerning neighborhoods with 2-sphere boundaries is necessary only for the points of $P(H)$.

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