the free loop space fibration is a locally trivial fiber bundle - reference?

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Let $Q$ be a compact Riemannian manifold. Then $\Lambda Q\rightarrow Q,$ $\gamma\mapsto \gamma(0)$ can be shown to be a locally trivial fiber bundle of Hilbert manifolds. Here, $\Lambda Q$ denotes the space of maps $S^1\rightarrow Q$ of Sobolev class $W^{1,2}.$

Question: Who proved it first? Is there an appropriate reference?

I once read it attributed to Klingenberg, but didn't find the proof (nor the statement) in the corresponding reference. I only know a proof due to Abbondandolo/Schwarz, but they claim no originality when asked.

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1 Answer

The proof isn't difficult, and doesn't depend on the class of maps, so I think it is one of those that is easier to write down a new proof than to trace it back in the literature.

If you want a reference to a published article containing a full proof, it is in my paper Constructing smooth manifolds of loop spaces as Corollary 4.8. This most certainly is not the first place that it appears (for one, I had a similar proof for the smooth case in my notes on the differential topology of loop spaces), but when I proved it then I did not rely on any other source. Also, the article version deals with a very wide range of classes of maps.

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