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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On homeomorphism spaces of Hilbert manifolds
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by Raymond Y. Wong PDF
Proc. Amer. Math. Soc. 89 (1983), 693-704 Request permission

Abstract:

Let $M$ be a Hilbert manifold modeled on the separable Hilbert space ${l^2}$. We prove the following Fibred Homeomorphism Extension Theorem: Let $M \times {B^n} \to {B^n}$ be the product bundle over the $n$-ball ${B^n}$, then any (fibred) homeomorphism on $M \times {S^{n - 1}}$ extends to a homeomorphism on all $M \times {B^n}$ if and only if it extends to a mapping on all $M \times {B^n}$. Moreover, the size of the extension may be restricted by any open cover on $M \times {B^n}$. The result is then applied to study the space of homeomorphisms $\mathcal {H}(M)$ under various topologies given on $\mathcal {H}(M)$. For instance, if $M = {l_2}$ and $\mathcal {H}({l_2})$ has the compact-open topology, the $\mathcal {H}({l_2})$ is an absolute extensor for all metric spaces. A counterexample is provided to show that the statement above may not be generalized to arbitrary manifold $M$.
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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 89 (1983), 693-704
  • MSC: Primary 57N20; Secondary 54C55, 58D05
  • DOI: https://doi.org/10.1090/S0002-9939-1983-0718999-2
  • MathSciNet review: 718999