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

 

Approximation by homeomorphisms and solution of P. Blass problem on pseudo-isotopy


Author: W. Holsztyński
Journal: Proc. Amer. Math. Soc. 27 (1971), 598-602
MSC: Primary 57.01
MathSciNet review: 0271949
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Abstract: For every map of $ f:{S^1} \to {S^1} = \{ z \in C:\vert z\vert = 1\} $ of degree 1, existence of apseudo-isotopy $ h:{S^1} \times I \to R = \{ z \in C:\vert z\vert \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}}$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0271949-X
PII: S 0002-9939(1971)0271949-X
Keywords: Pseudo-isotopy, generalizated pseudo-isotopy, isotopy domination, approximation by homeomorphisms, maps of degree 1, sphere $ {S^1}$ and $ {S^n}$, complex plane $ C$, Euclidean space $ {E^{n + 1}}$
Article copyright: © Copyright 1971 American Mathematical Society