Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 54.78

Retrieve articles in all journals with MSC: 54.78


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1971-0275401-1
PII: S 0002-9947(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