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

On homeomorphism spaces of Hilbert manifolds


Author: Raymond Y. Wong
Journal: Proc. Amer. Math. Soc. 89 (1983), 693-704
MSC: Primary 57N20; Secondary 54C55, 58D05
MathSciNet review: 718999
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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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DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0718999-2
PII: S 0002-9939(1983)0718999-2
Keywords: Hilbert manifolds, spaces of homeomorphisms, absolute neighborhood extensors, $ {\mathbf{Z}}$-sets
Article copyright: © Copyright 1983 American Mathematical Society