Timeline for Important results that use infinite-dimensional manifolds?

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May 13, 2016 at 14:38	history	made wiki			Post Made Community Wiki by Todd Trimble
Jan 27, 2010 at 23:12	comment	added	Ryan Budney		Fadell and Neuwirth's argument was a finite-dimensional one but it was inspired by Palais's result that if $N$ is a submanifold of $M$ then the restriction map $Emb(M, X) \to Emb(N, X)$ is a locally trivial fibre bundle, where $Emb(M,X)$ denotes the space of smooth embeddings of $M$ in $X$. The Fadell-Neuwirth argument is for the special situation where $M$ and $N$ are finite sets.
Jan 27, 2010 at 22:59	comment	added	algori		Ryan -- where does the proof of the fact that pure braid groups are iterated semi-direct products of free groups use infinite-dimensional manifolds? This follows simply from the fact that the ordered configuration space $F(\mathbf{R}^2,k)$ of $k$ distinct points in the plane is fibered over $F(\mathbf{R}^2,k-1)$ with fiber homotopy equivalent to the wedge of $k-1$ circles.
Jan 4, 2010 at 17:38	comment	added	Mark Meckes		Your summary can be paraphrased to answer the more basic question of why infinite dimensional vector spaces are important: even if your primary interest is in things happening in finite dimensional spaces (e.g. PDEs), then infinite dimensional spaces of functions come up naturally.
Jan 1, 2010 at 20:03	history	edited	Ryan Budney	CC BY-SA 2.5	more examples
Dec 31, 2009 at 1:47	history	edited	Ryan Budney	CC BY-SA 2.5	elaboration
Dec 31, 2009 at 1:42	history	answered	Ryan Budney	CC BY-SA 2.5