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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Decomposition spaces having arbitrarily small neighborhoods with $ 2$-sphere boundaries


Author: Edythe P. Woodruff
Journal: Trans. Amer. Math. Soc. 232 (1977), 195-204
MSC: Primary 57A10; Secondary 54B15
MathSciNet review: 0442944
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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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DOI: http://dx.doi.org/10.1090/S0002-9947-1977-0442944-2
PII: S 0002-9947(1977)0442944-2
Article copyright: © Copyright 1977 American Mathematical Society