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

 

Somewhere locally flat codimension one manifolds with $ 1-{\rm ULC}$ complements are locally flat


Authors: T. M. Price and C. L. Seebeck
Journal: Trans. Amer. Math. Soc. 193 (1974), 111-122
MSC: Primary 57A45
MathSciNet review: 0346796
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Abstract: The purpose of this paper is to prove a taming theorem for a codimension one manifold that is locally flat at some point and has 1-ULC complement. We also prove that any two sufficiently close locally flat embeddings of a codimension one manifold are ambient isotopic. Since this paper was first submitted, R. Daverman has shown that, given any point on a codimension one manifold with 1-ULC complement, some neighborhood of that point lies on a codimension one sphere that is locally flat at some points and has 1-ULC complement. Hence the two papers combined prove that a codimension one manifold is locally flat if and only if its complement is 1-ULC.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0346796-8
PII: S 0002-9947(1974)0346796-8
Keywords: Codimension one, homeomorphic approximation, 1-ULC complement, radial engulfing
Article copyright: © Copyright 1974 American Mathematical Society