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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Embedding phenomena based upon decomposition theory: locally spherical but wild codimension one spheres
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by Robert J. Daverman PDF
Proc. Amer. Math. Soc. 90 (1984), 139-144 Request permission

Abstract:

For $n \geqslant 7$ we describe an $(n - 1)$-sphere $\Sigma$ wildly embedded in the $n$-sphere yet every point of $\Sigma$ has arbitrarily small neighborhoods bounded by flat $(n - 1)$-spheres, each intersecting $\Sigma$ in an $(n - 2)$-sphere. Not only do these examples for large $n$ run counter to what can occur when $n = 3$, they also illustrate the sharpness of high-dimensional taming theorems developed by Cannon and Harrold and Seebeck. Furthermore, despite their wildness, they have mapping cylinder neighborhoods, which both run counter to what is possible when $n = 3$ and also partially illustrate the sharpness of another high-dimensional taming theorem due to Bryant and Lacher.
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Additional Information
  • © Copyright 1984 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 90 (1984), 139-144
  • MSC: Primary 57N50; Secondary 54B15, 57M30, 57N15, 57N45
  • DOI: https://doi.org/10.1090/S0002-9939-1984-0722432-5
  • MathSciNet review: 722432