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 ${S^3}$.References
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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