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Transactions of the American Mathematical Society

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



Almost locally tame $2$-manifolds in a $3$-manifold

Author: Harvey Rosen
Journal: Trans. Amer. Math. Soc. 156 (1971), 59-71
MSC: Primary 54.78
MathSciNet review: 0275401
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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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Keywords: Almost locally tame 2-manifolds, 2-manifolds in 3-manifolds, tameness from a complementary domain, wildness from a complementary domain, locally peripherally collared 2-manifolds, convergent sequence of 2-manifolds, locally spanned 2-manifolds, piercing disk, almost locally polyhedral tori, complements of tame knots
Article copyright: © Copyright 1971 American Mathematical Society