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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
DOI: https://doi.org/10.1090/S0002-9947-1974-0346796-8
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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  • [1] J. L. Bryant, Euclidean n-space modulo an $ (n - 1)$-cell, Trans. Amer. Math. Soc. 179 (1973), 181-192. MR 0324703 (48:3053)
  • [2] A. V. Černavskiĭ, The k-stability of homeomorphisms and the union of cells, Dokl. Akad. Nauk SSSR 180 (1968), 1045-1047 = Soviet Math. Dokl. 9 (1968), 729-732. MR 37 #6919. MR 0231364 (37:6919)
  • [3] -, Locally homotopic unknotted imbeddings of manifolds, Dokl. Akad. Nauk SSSR 181 (1968), 290-293 = Soviet Math. Dokl. 9 (1968), 835-839. MR 38 #720. MR 0232395 (38:720)
  • [4] R. D. Edwards and R. C. Kirby, Deformations of spaces of imbeddings, Ann. of Math. (2) 93 (1971), 63-88. MR 44 #1032. MR 0283802 (44:1032)
  • [5] R. C. Kirby and L. C. Siebenmann, On the triangulation of manifolds and the hauptvermutung, Bull. Amer. Math. Soc. 75 (1969), 742-749. MR 39 #3500. MR 0242166 (39:3500)
  • [6] C. L. Seebeck III, Collaring an $ (n - 1)$-manifold in an n-manifold, Trans. Amer. Math. Soc. 148 (1970), 63-68. MR 41 #2692. MR 0258045 (41:2692)
  • [7] P. Wright, A uniform generalized Schoenflies theorem, Ann. of Math. (2) 89 (1969), 292-304. MR 40 #3556. MR 0250317 (40:3556)

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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

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