Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

A new characterization of tame $ 2$-spheres in $ E\sp{3}$


Author: Lawrence R. Weill
Journal: Trans. Amer. Math. Soc. 190 (1974), 243-252
MSC: Primary 57A50; Secondary 57A10
DOI: https://doi.org/10.1090/S0002-9947-1974-0339188-9
MathSciNet review: 0339188
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown in Theorem 1 that a 2-sphere S in $ {E^3}$ is tame from $ A = {\text{Int}}\;S$ if and only if for each compact set $ F \subset A$ there exists a 2-sphere $ S'$ with complementary domains $ A' = {\text{Int}}\;S',B' = {\text{Ext}}\;S'$, such that $ F \subset A' \subset \overline {A'} \subset A$ and for each $ x \in S'$ there exists a path in $ \overline {B'} $ of diameter less than $ \rho (F,S)$ which runs from x to a point $ y \in S$. Furthermore, the theorem holds when A is replaced by B, $ A'$ by $ B',B'$ by $ A'$, and Int by Ext. Two applications of this characterization are given. Theorem 2 states that a 2-sphere is tame from the complementary domain C if for arbitrarily small $ \varepsilon > 0$, S has a metric $ \varepsilon $-envelope in C which is a 2-sphere. Theorem 3 answers affirmatively the following question: Is a 2-sphere $ S \subset {E^3}$ tame in $ {E^3}$ if there exists an $ \varepsilon > 0$ such that if a, $ b \in S$ satisfy $ \rho (a,b) < \varepsilon $, then there exists a path in S of spherical diameter $ \rho (a,b)$ which connects a and b?


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

  • [1] R. H. Bing, A surface is tame if its complement is 1-ULC, Trans. Amer. Math. Soc. 101 (1961), 294-305. MR 24 #A1117. MR 0131265 (24:A1117)
  • [2] -, Approximating spheres with polyhedral ones, Ann. of Math. (2) 65 (1967), 456-483.
  • [3] C. E. Burgess and J. W. Cannon, Embeddings of surfaces in $ {E^3}$, Rocky Mountain J. Math. 1 (1971), no. 2, 259-344. MR 43 #4008. MR 0278277 (43:4008)
  • [4] J. W. Cannon, $ \ast $-taming sets for crumpled cubes. II: Horizontal sections in closed sets, Trans. Amer. Math. Soc. 161 (1971), 441-446. MR 43 #8066. MR 0282354 (43:8066)
  • [5] John Cobb, Visible taming of 2-spheres in $ {E^3}$ (manuscript).
  • [6] L. D. Loveland, A surface in $ {E^3}$ is tame if it has round tangent balls, Trans. Amer. Math. Soc. 152 (1970), 389-397. MR 42 #5270. MR 0270381 (42:5270)
  • [7] L. R. Weill, A new characterization of tame 2-spheres in $ {E^3}$, Ph. D. dissertation, University of Idaho, Moscow, Id., June, 1971.
  • [8] R. L. Wilder, Topology of manifolds, Amer. Math. Soc. Colloq. Publ., vol. 32, Amer. Math. Soc., Providence, R. I., 1963. MR 32 #440. MR 0182958 (32:440)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57A50, 57A10

Retrieve articles in all journals with MSC: 57A50, 57A10


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0339188-9
Keywords: Tame surfaces, tame 2-spheres, surfaces in $ {E^{3}}$, characterizations of tameness
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society