I'm looking for the origins of the isotopy extension theorem in categories other than the smooth category.

Precisely, in the smooth category, the isotopy extension theorem says that if $f : [0,1] \times M \to N$ is a smooth 1-parameter family of embeddings (with $N$ boundaryless, and $M$ compact) then there exists $F : [0,1] \times N \to N$ a smooth $1$-parameter family of diffeomorphisms so that $F(0, \cdot) = Id_N$ and $F(t,f(0,x)) = f(t,x)$ for all $(t,x) \in [0,1] \times M$. This was seen to be a "very natural" theorem by Palais, with the generalization stating that the restriction map $Diff(N) \to Emb(M,N)$ was not only a Serre fibration but a locally trivial fibre bundle. In this ideal case, the references are:

  • Palais, Richard S. Local triviality of the restriction map for embeddings. Comment. Math. Helv. 34 1960 305–312.

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

Let $Aut(N)$ be the automorphisms of the manifold $N$ in whichever category of manifolds it lives in (topological, $PL$ or smooth). My understanding is its known that the restriction map

$$Aut(N) \to Emb(M,N)$$

is known to be a Serre fibration provided $N$ is a co-dimension $0$ submanifold of $N$, in any of the above three manifold categories.

My questions:

Q1: Where were these results first proven in the PL and TOP cases? Are they known for all dimensions?

Q2: If one allows $M$ to have co-dimension $> 0$, what is known about this map being or not being a fibration?

I'm in the process of trying to both learn the basics and get an overview of smoothing theory. Any help is appreciated.

  • $\begingroup$ By $F(t,f(0,x)) = F(t,x)$, you most probably mean $F(t,f(0,x)) = f(t,x)$. Do you know any reference for an orbifold version of this theorem? $\endgroup$ – Mohammad Farajzadeh-Tehrani Feb 23 '16 at 19:39

The original reference for the topological isotopy extension theorem is Corollary 1.4 of

Edwards, Robert D.; Kirby, Robion C. Deformations of spaces of imbeddings. Ann. Math. (2) 93 1971 63–88.

Note that the isotopy is assumed be locally flat, and $N$ and $M$ can have boundaries as long as they are preserved (the word proper in the statement refers to the condition $h^{-1}(\partial N)=\partial M$).

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    $\begingroup$ Thanks! So if there's a Palais-style variant of this theorem, it must occur in a year $\geq 1971$. :) $\endgroup$ – Ryan Budney Oct 23 '13 at 15:41

It looks like the Palais-style variant of the PL isotopy extension theorem is proven in:

  • Hudson, John F. P. Extending piecewise-linear isotopies. Proc. London Math. Soc. (3) 16 1966 651–668.

In this article Hudson says the result is first proven in M.C. Irwin's dissertation at Cambridge.

By this I mean he's showing $Aut(N) \to Emb(M,N)$ is a Serre fibration. He also assumes co-dimension $\geq 3$.

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