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I'm looking for a reference or a proof of the following statement :

Let $M$ be a compact smooth manifold with boundary. Then the space of embeddings $\partial M\times[0,1]\to M$ inducing the identity $\partial M\times\{0\}\to \partial M$ is contractible.

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    $\begingroup$ What topology do you want to use on that space? When your domain is non-compact there's a lot of different topologies to choose from. If you let the domain be $\partial M \times [0,1]$ (being the identity on $M \times \{0\}$ then you've got a more-or-less canonical topology on the space. People call it the weak or Whitney topology. $\endgroup$
    – Ryan Budney
    Commented Oct 12, 2012 at 0:05
  • $\begingroup$ Ryan. I'm not sure I have some flexibility on what topology I can use. I have edited the question to take your comment into account. $\endgroup$
    – Geoffroy Horel
    Commented Oct 12, 2012 at 0:18
  • $\begingroup$ You still have flexibility. There's the $C^0$ uniform topology, the $C^1$ uniform topology, etc, up to the $C^\infty$ weak/Whitney topology. $\endgroup$
    – Ryan Budney
    Commented Oct 12, 2012 at 0:39

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The theorem you're looking for is proven in Cerf's dissertation.

J. Cerf, Topologie de certains espaces de plongements, Bull S.M.F., tome 89 (1961) 227-380.

This is for the case you mention, when you look at the space of $C^k$-smooth embeddings with the $C^k$-Whitney/weak topology. Cerf proves a lot of other similar foundational results about embedding spaces, tubular neighbourhoods, collars, restriction maps and so on.

To set up the proof, just take your favourite proof that collars are unique up to isotopy. These proofs can all be souped-up to work at the embedding space level, you just have to make smart choices for your maps.

For example, consider the space of smooth embeddings $f : [0,1] \to [0,\infty)$ with $f(0)=0$. Here is a homotopy:

$$F(t,x) = \frac{f((1-t)x)}{1-t} \text{ for } t < 1$$

and

$$F(1,x) = f'(0)x$$

Notice that this is an isotopy from $f$ to a linear map. The space of linear embeddings is contractible.

The proof for general collar neighbourhoods is not much more sophisticated than the above, but you have the complication of not having such nice coordinates in your manifold. You can make up for that in various ways, one being embedding the manifold in Euclidean space and using tubular neighbourhoods. Another would be to use flows.

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