6
$\begingroup$

Let $M^n$ be a smooth $n$-dimensional manifold and let $\mathbb{D}^n$ be the open unit disk in $\mathbb{R}^n$. Consider the space $\text{Emb}(\mathbb{D}^n,M^n)$ of smooth embeddings of $\mathbb{D}^n$ into $M^n$, equipped with the $C^{\infty}$ topology. Letting $\text{Fr}(M^n)$ be the bundle of frames of the tangent bundle of $M^n$, there is a map $\Psi\colon \text{Emb}(\mathbb{D}^n,M^n) \rightarrow \text{Fr}(M^n)$ taking an embedding $f\colon \mathbb{D}^n \rightarrow M^n$ to the point $f(0) \in M^n$ equipped with the image of the standard framing of $T_0 \mathbb{D}^n = \mathbb{R}^n$ under the derivative $D_0 f\colon T_0 \mathbb{D}^n \rightarrow T_{f(0)} M^n$.

For a paper I'm writing, I'm looking for a reference proving that $\Psi$ is a homotopy equivalence. I know several proofs of this, but all of them have slightly annoying aspects (at least if you write out all the details) so I would prefer to just give a reference.

$\endgroup$
4
  • $\begingroup$ There is the lecture notes, Diffeomorphism groups of manifolds by Alexander Kupers $\endgroup$ May 19, 2022 at 16:49
  • $\begingroup$ @DanielH.Hartman: The version of that I have is about 330 pages. Do you know where this result can be found in it? $\endgroup$ May 19, 2022 at 17:26
  • 1
    $\begingroup$ chapter 9, page 73. $\endgroup$ May 19, 2022 at 18:05
  • $\begingroup$ @DanielH.Hartman: Thanks! $\endgroup$ May 19, 2022 at 23:27

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.