# Are there non-trivial locally trivial fiber bundles of Hilbert manifolds if the fiber is infinite-dimensional?

It is well-known (see Kuiper's theorem) that every Hilbert space bundle over a manifold is trivial if the Hilbert space is truly infinite-dimensional.

Does the same hold true if one considers locally trivial fiber bundles instead and the fiber is assumed to be a truly infinite dimensional Hilbert manifold?

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What do you mean by "truly" infinite dimensional Hilbert manifold? You can easily build a fiber bundle over $S^1$ with fiber $S^1 \times H$, $H$ a Hilbert space, which is non-trivial just by homotopy reasons. – Alberto Abbondandolo May 2 '12 at 13:08