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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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Almost locally tame $2$-manifolds in a $3$-manifold
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by Harvey Rosen PDF
Trans. Amer. Math. Soc. 156 (1971), 59-71 Request permission

Abstract:

Several conditions are given which together imply that a 2-manifold M in a 3-manifold is locally tame from one of its complementary domains, U, at all except possibly one point. One of these conditions is that certain arbitrarily small simple closed curves on M can be collared from U. Another condition is that there exists a certain sequence ${M_1},{M_2}, \ldots$ of 2-manifolds in U converging to M with the property that each unknotted, sufficiently small simple closed curve on each ${M_i}$ is nullhomologous on ${M_i}$. Moreover, if each of these simple closed curves bounds a disk on a member of the sequence, then it is shown that M is tame from $U(M \ne {S^2})$. As a result, if U is the complementary domain of a torus in ${S^3}$ that is wild from U at just one point, then U is not homeomorphic to the complement of a tame knot in ${S^3}$.
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Additional Information
  • © Copyright 1971 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 156 (1971), 59-71
  • MSC: Primary 54.78
  • DOI: https://doi.org/10.1090/S0002-9947-1971-0275401-1
  • MathSciNet review: 0275401