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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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On embeddings with locally nice cross-sections
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by J. L. Bryant PDF
Trans. Amer. Math. Soc. 155 (1971), 327-332 Request permission

Abstract:

A $k$-dimensional compactum ${X^k}$ in euclidean space ${E^n}(n - k \geqq 3)$ is said to be locally nice in ${E^n}$ if ${E^n} - {X^k}$ is $1$-ULC. In this paper we prove a general theorem which implies, in particular, that ${X^k}$ is locally nice in ${E^n}$ if the intersection of ${X^k}$ with each horizontal hyperplane of ${E^n}$ is locally nice in the hyperplane. From known results we obtain immediately that a $k$-dimensional polyhedron $P$ in ${E^n}$ ($n - k \geqq 3$ and $n \geqq 5$) is tame in ${E^n}$ if each $({E^{n - 1}} \times \{ w\} ) - P(w \in {E^1})$ is $1$-ULC. However, by strengthening our general theorem in the case $n = 4$, we are able to prove this result for $n = 4$ as well. For example, an arc $A$ in ${E^4}$ is tame if each horizontal cross-section of $A$ is tame in the cross-sectional hyperplane (that is, lies in an arc that is tame in the hyperplane).
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Additional Information
  • © Copyright 1971 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 155 (1971), 327-332
  • MSC: Primary 57.05
  • DOI: https://doi.org/10.1090/S0002-9947-1971-0276983-6
  • MathSciNet review: 0276983