Contractibility of the space of collars - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T19:38:43Z http://mathoverflow.net/feeds/question/109422 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109422/contractibility-of-the-space-of-collars Contractibility of the space of collars Geoffroy Horel 2012-10-11T23:23:49Z 2012-10-12T01:08:38Z <p>I'm looking for a reference or a proof of the following statement :</p> <p>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.</p> http://mathoverflow.net/questions/109422/contractibility-of-the-space-of-collars/109424#109424 Answer by Ryan Budney for Contractibility of the space of collars Ryan Budney 2012-10-12T00:52:35Z 2012-10-12T00:52:35Z <p>The theorem you're looking for is proven in Cerf's dissertation. </p> <p>J. Cerf, Topologie de certains espaces de plongements, Bull S.M.F., tome 89 (1961) 227-380. </p> <p>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. </p> <p>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. </p> <p>For example, consider the space of smooth embeddings $f : [0,1] \to [0,\infty)$ with $f(0)=0$. Here is a homotopy:</p> <p>$$F(t,x) = \frac{f((1-t)x)}{1-t} \text{ for } t &lt; 1$$</p> <p>and</p> <p>$$F(1,x) = f'(0)x$$</p> <p>Notice that this is an isotopy from $f$ to a linear map. The space of linear embeddings is contractible. </p> <p>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. </p>