On homeomorphism spaces of Hilbert manifolds

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 $M$.