the free loop space fibration is a locally trivial fiber bundle - reference? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T14:05:56Z http://mathoverflow.net/feeds/question/63323 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/63323/the-free-loop-space-fibration-is-a-locally-trivial-fiber-bundle-reference the free loop space fibration is a locally trivial fiber bundle - reference? Orbicular 2011-04-28T18:44:40Z 2011-11-18T19:22:12Z <p>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 $W^{1,2}.$</p> <p>Question: Who proved it first? Is there an appropriate reference?</p> <p>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.</p> http://mathoverflow.net/questions/63323/the-free-loop-space-fibration-is-a-locally-trivial-fiber-bundle-reference/63337#63337 Answer by Andrew Stacey for the free loop space fibration is a locally trivial fiber bundle - reference? Andrew Stacey 2011-04-28T20:58:22Z 2011-04-28T20:58:22Z <p>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.</p> <p>If you want a reference to a published article containing a full proof, it is in my paper <a href="http://www.math.ntnu.no/~stacey/Research/Papers/smooth.html" rel="nofollow">Constructing smooth manifolds of loop spaces</a> 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.</p>