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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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Keywords: Hilbert manifolds, spaces of homeomorphisms, absolute neighborhood extensors, <!– MATH ${\mathbf {Z}}$ –> <IMG WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0" SRC="images/img20.gif" ALT="${\mathbf {Z}}$">-sets
Article copyright: © Copyright 1983 American Mathematical Society