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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Slicing theorems for $n$-spheres in Euclidean $(n+1)$-space
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by Robert J. Daverman PDF
Trans. Amer. Math. Soc. 166 (1972), 479-489 Request permission

Abstract:

This paper describes conditions on the intersection of an n-sphere $\Sigma$ in Euclidean $(n + 1)$-space ${E^{n + 1}}$ with the horizontal hyperplanes of ${E^{n + 1}}$ sufficient to determine that the sphere be nicely embedded. The results generally are pointed towards showing that the complement of $\Sigma$ is 1-ULC (uniformly locally 1-connected) rather than towards establishing the stronger property that $\Sigma$ is locally flat. For instance, the main theorem indicates that ${E^{n + 1}} - \Sigma$ is 1-ULC provided each non-degenerate intersection of $\Sigma$ and a horizontal hyperplane be an $(n - 1)$-sphere bicollared both in that hyperplane and in $\Sigma$ itself $(n \ne 4)$.
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Additional Information
  • © Copyright 1972 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 166 (1972), 479-489
  • MSC: Primary 57A35
  • DOI: https://doi.org/10.1090/S0002-9947-1972-0295356-4
  • MathSciNet review: 0295356