Approximation by homeomorphisms and solution of P. Blass problem on pseudo-isotopy
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- by W. Holsztyński
- Proc. Amer. Math. Soc. 27 (1971), 598-602
- DOI: https://doi.org/10.1090/S0002-9939-1971-0271949-X
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Abstract:
For every map of $f:{S^1} \to {S^1} = \{ z \in C:|z| = 1\}$ of degree 1, existence of apseudo-isotopy $h:{S^1} \times I \to R = \{ z \in C:|z| \geqq 1\}$ such that $h(z,0) = z$ and $h(z,1) = f(z)$ is established. On the other hand (i) maps of ${I^n}$ into ${I^n} \times 0 \subset {E^{n + 1}}$ cannot be, in general, uniformly approximated by homeomorphic embeddings of ${I^n}$ in ${E^{n + 1}}$ for $n > 1$, and (ii) maps of ${S^n}$ into ${S^n} \subset {E^n}$ of degree 1 cannot be, in general, extended to a pseudo-isotopy of ${S^n}$ into ${E^{n + 1}}$.References
- Karol Borsuk, Theory of retracts, Monografie Matematyczne, Tom 44, Państwowe Wydawnictwo Naukowe, Warsaw, 1967. MR 0216473
- V. Gol′shtynskiĭ and S. Iliadis, Approximation of multi-valued by single-valued mappings and some applications, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 16 (1968), 765–769 (Russian, with English summary). MR 239572
Bibliographic Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 27 (1971), 598-602
- MSC: Primary 57.01
- DOI: https://doi.org/10.1090/S0002-9939-1971-0271949-X
- MathSciNet review: 0271949