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$\DeclareMathOperator\Diff{Diff}\DeclareMathOperator\Imb{Imb}$Notation: Let $M$ be a smooth, closed manifold, $S$ any submanifold of $M$, $\Diff(M)$ the group of diffeomorphisms of $M$ and $\Imb(S, M)$ the space of smooth imbeddings of $S$ into $M$.

A classical result of R. Palais from the 1960 paper Local triviality of the restriction map for embeddings says that the map $\Diff(M)\rightarrow\Imb(S, M)$ given by restriction is a fibration.

I feel like I've heard during numerous teas that there are various refinements and generalizations of this due to J. Cerf and (possibly) others.

(1) Can anyone summarize what else is known in this direction beyond the theorem of Palais?

(2) Is there a way to see Palais' result easily? [added: from the responses it sounds like the original paper is still a great way to see this result. But see the answers of Randal-Williams and Palais for an alternate route.]

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    $\begingroup$ My article, appeared in Commentarii Mathematici Helvetici with the title "Local triviality of the restriction map for embeddings". Elon Lima's article with a shorter proof of my main result has the same title with an "On" in front and appears in a later issue of the same journal. If you have a good feeling for basic differential topology the proof is quite natural and non-technical. I recall it originated in a discussion I had with Steve Smale. We decided that the restriction map should be a Serre fibration. I got interested in it and we were both surprised that it was locally trivial. $\endgroup$
    – Dick Palais
    Commented Sep 11, 2010 at 15:56
  • $\begingroup$ Thanks for the specific reference as well as the historical anecdote! $\endgroup$
    – Romeo
    Commented Sep 11, 2010 at 17:01
  • $\begingroup$ I apologise for the trivial question, but what is the group structure on $Imb(S,M)$ mentioned in the first sentence of the question? $\endgroup$
    – Alex M.
    Commented Apr 12, 2014 at 7:58
  • $\begingroup$ @AlexM.: That's just a typo. Embeddings is just a space, and generally not a group. $\endgroup$
    – Ryan Budney
    Commented Aug 15, 2023 at 6:32

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It's not clear what you mean by "various refinements and generalizations". Cerf has a huge paper published by IHES "Topologie de certains espaces de plongements" which goes into many related details. In a way it's more of a ground-up collection of basic information on the topology of function spaces.

Regarding your 2nd question, if instead of demanding a fibre bundle you ask for a Serre fibration, the proof is relatively simple. It's just the isotopy extension theorem with parameters, and the proof is pretty much verbatim Hirsch's proof of isotopy extension in his "Differential Topology" text plus the observation that solutions depend smoothly on the initial conditions.

Regarding your 2nd question, yes of course. Palais's paper is quite nice. If you haven't had a look at it, you might as well try -- it's only 7 pages long. If you want to discover the proof on your own I'd start with the case $S$ a finite set. Then move up to $S$ a positive-dimensional submanifold. You'll want to be comfortable with things like the proof of the tubular neighbourhood theorem, the concept of injectivity radius, etc.

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    $\begingroup$ Ryan, which paper of Palais is the OP referring to? Thanks in advance for the reference. $\endgroup$
    – Willie Wong
    Commented Sep 11, 2010 at 11:12
  • $\begingroup$ Appended to the original post. $\endgroup$
    – Romeo
    Commented Sep 11, 2010 at 17:01
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This is closely related to (and can be proved by the same methods as) the fact that if we fix another manifold $N$ then the restriction map $$\mathrm{Emb}(M, N) \to \mathrm{Emb}(S, N)$$ is a locally trivial fibration, which was proved in the same paper by Palais. However, I wanted to advertise a geodesic proof of this fact:

E. L. Lima, On the local triviality of the restriction map for embeddings, Commentarii Mathematici Helvetici Volume 38, Number 1, pp 163-164.

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