Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Extension d'un théorème de Louis Antoine


Author: Nikias Stavroulakis
Journal: Proc. Amer. Math. Soc. 39 (1973), 201-210
MSC: Primary 55A25
DOI: https://doi.org/10.1090/S0002-9939-1973-0317313-8
MathSciNet review: 0317313
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ f:[0,1] \times S \to {R^3}$ be a map subject to the conditions: (1) $ f\vert]0,1] \times S$ is $ (1,1)$ into; (2) $ f\vert\{ 0\} \times S$ is not $ (1,1)$ into; (3) The image $ {\Gamma _0} = f(\{ 0\} \times S)$ is a Jordan curve ; (4) $ f(\{ 0\} \times S) \cap f(]0,1] \times S) = \emptyset $.

Let $ \mu $ be the linking number of each of the curves $ {\Gamma _t} = f(\{ t\} \times S),t \in ]0,1]$, with $ {\Gamma _0}$. Let $ v$ be the degree of the mapping $ h:S \to {\Gamma _0}$ defined by $ h(u) = f(0,u)$. We prove that, if $ {\Gamma _0}$ is tame, the integers $ \mu $ and $ v$ are relatively prime. The question is open in case that $ {\Gamma _0}$ is wild.


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

  • [1] L. Antoine, Sur l'homéomorphie de deux figures et de leurs voisinages, J. Math. Pures Appl. (8) 4 (1921), 221-325.
  • [2] R. H. Bing, Approximating surfaces with polyhedral ones, Ann. of Math. (2) 65 (1957), 465–483. MR 0087090

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 55A25

Retrieve articles in all journals with MSC: 55A25


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0317313-8
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society