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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The canonical Schoenflies theorem


Author: David B. Gauld
Journal: Proc. Amer. Math. Soc. 27 (1971), 603-612
MSC: Primary 54.78
MathSciNet review: 0279786
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Abstract: M. Brown has shown that if $ f:{S^{n - 1}} \times I' \to {S^n}$, where $ I' = [ - 1,1]$, is an embedding, then the closure of either complementary domain of $ f({S^{n - 1}} \times 0)$ is homeomorphic to $ {B^n}$; in fact there is an embedding $ g : {B^n} \to {S^n}$ satisfying $ g\vert{S^{n - 1}} = f\vert{S^{n - 1}} \times 0$. This paper shows that the choice of embedding $ g$ can be made to be ``canonical,'' i.e. if $ f'$ is an embedding near $ f$ in the compact-open sense, then the embedding $ g'$ corresponding to $ f'$ is near $ g$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0279786-7
PII: S 0002-9939(1971)0279786-7
Keywords: Embedding spheres in spheres, locally-flat embeddings, Schoenflies theorem
Article copyright: © Copyright 1971 American Mathematical Society