Somewhere locally flat codimension one manifolds with 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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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1974-0346796-8

Keywords:
Codimension one,
homeomorphic approximation,
1-ULC complement,
radial engulfing

Article copyright:
© Copyright 1974
American Mathematical Society