# Isotopy extension theorems

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.

• 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? – 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 $h^{-1}(\partial N)=\partial M$).

• Thanks! So if there's a Palais-style variant of this theorem, it must occur in a year $\geq 1971$. :) – 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 $\geq 3$.