The canonical Schoenflies theorem
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- by David B. Gauld
- Proc. Amer. Math. Soc. 27 (1971), 603-612
- DOI: https://doi.org/10.1090/S0002-9939-1971-0279786-7
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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 $g|{S^{n - 1}} = f|{S^{n - 1}} \times 0$. 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 $g$.References
- Marston Morse, A reduction of the Schoenflies extension problem, Bull. Amer. Math. Soc. 66 (1960), 113–115. MR 117694, DOI 10.1090/S0002-9904-1960-10420-X
- William Huebsch and Marston Morse, The dependence of the Schoenflies extension on an accessory parameter (the topological case), Proc. Nat. Acad. Sci. U.S.A. 50 (1963), 1036–1037. MR 182957, DOI 10.1073/pnas.50.6.1036
- J. M. Kister, Microbundles are fibre bundles, Ann. of Math. (2) 80 (1964), 190–199. MR 180986, DOI 10.2307/1970498
Bibliographic Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 27 (1971), 603-612
- MSC: Primary 54.78
- DOI: https://doi.org/10.1090/S0002-9939-1971-0279786-7
- MathSciNet review: 0279786