On homeomorphism spaces of Hilbert manifolds
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 by Raymond Y. Wong PDF
 Proc. Amer. Math. Soc. 89 (1983), 693704 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 compactopen 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 $M$.References

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Additional Information
 © Copyright 1983 American Mathematical Society
 Journal: Proc. Amer. Math. Soc. 89 (1983), 693704
 MSC: Primary 57N20; Secondary 54C55, 58D05
 DOI: https://doi.org/10.1090/S00029939198307189992
 MathSciNet review: 718999