Almost locally tame -manifolds in a -manifold

Author:
Harvey Rosen

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

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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 of 2-manifolds in *U* converging to *M* with the property that each unknotted, sufficiently small simple closed curve on each is nullhomologous on . 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 . As a result, if *U* is the complementary domain of a torus in that is wild from *U* at just one point, then *U* is not homeomorphic to the complement of a tame knot in .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1971-0275401-1

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