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Suppose we have a diffeomorphism $f:{\mathbb{S}}^n_{+}\to\mathbb{S}^n$ of class $C^1$ of the closed upper hemisphere onto a submanifold of $\mathbb{S}^n$ with boundary.

Question. Is it possible to extend it to a diffeomorphism $F:\mathbb{S}^n\to\mathbb{S}^n$?

I believe that when $n=2$ this should follow from the smooth Schoenflies theorem, but still some work is necessary. I think I know how to do, but I did not write a rigorous proof.

From what I understand when $n\geq 3$, the generalized Schoenflies theorem allows us to extend $f$ to a homeomorphism, but that is much less than extending to a diffeomorphism.

I expect that the answer might depend on $n$ ($n\geq 3$, $n=4$, $n\geq 5$).

I would greatly appreciate it if you could provide references where I cold find relevant theorems (if there are any), including the case $n=2$. I need references for a proper citation in my paper.

Edit: This question is strictly related to another post: Gluing two diffeomorphisms together

Piotr Hajlasz
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    I think the issue in Schoenflies is whether a smoothly embedded codimension one sphere bounds a ball. Your setting is easier because you have an embedded disk. I think any two smoothly embedded codimension zero disks are equivalent up to ambient diffeomorphism by a result of Palais in "Extending diffeomorphisms", https://www.ams.org/journals/proc/1960-011-02/S0002-9939-1960-0117741-0/S0002-9939-1960-0117741-0.pdf. – Igor Belegradek Jan 29 '23 at 19:19
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    By Palais, you can assume that $F$ maps hemisphere to itself. Then in dimension $n<7$ an extension always exists. In higher dimensions, it depends on the number of exotic spheres. – Moishe Kohan Jan 29 '23 at 19:31
  • @MoisheKohan: I think your comment refers to the restriction map $r: Diff(D^n)\to Diff(S^{n-1})$ which is a fibration whose homotopy fiber is $Diff(D^n, rel \partial)$. Look at the $\pi_0$ portion of the homotopy exact sequence of the fibration. We are interested in whether the rightmost arrow $r_$ is onto. If $n>4$, the kernel of $r$ is the group of homotopy $(n+1)$-spheres. What do homotopy spheres have to do with surjectivity of $r_$? Could you elaborate? – Igor Belegradek Jan 29 '23 at 20:04
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    @MoisheKohan I think Palais' theorem says exactly that you can always extend, because (if I correctly understand his paper) you can compose $f$ with a diffeomorphism of $\mathbb{S}^n$ so that $f$ becomes identity. I am confused. – Piotr Hajlasz Jan 29 '23 at 20:08
  • Oh, I see: the sequence extends to the right and the next term is $\pi_0(BDiff(D^n, rel\partial))$ which is always trivial (contactibility of $EG$ implies path-connectedness of $BG$). Thus it looks like $r_*$ is onto on $\pi_0$, and hence by the homotopy lifting property one can always extend. – Igor Belegradek Jan 29 '23 at 20:23
  • @PiotrHajlasz: I think just like you explain everything follows from Palais. The only technical issue is regularity at the equator. If a diffeomorphism of the upper hemisphere is identity on the equator, then its extension by the identity on the lower hemisphere need not be smooth on the ambient sphere. What one has to do is to first deform to a diffeomorphism of the upper hemisphere that is identity near the boundary. – Igor Belegradek Jan 29 '23 at 20:45
  • Yes. You are right, I was overthinking it. – Moishe Kohan Jan 29 '23 at 21:06
  • @IgorBelegradek I still need to complete reading of Palais' paper, but I don't think the boundary is an issue since Palais' result deals with diffeomorphism up to the bounday so we can assume that one of them is the identity on the closed hemisphere. – Piotr Hajlasz Jan 29 '23 at 21:22
  • @PiotrHajlasz: yes, you are right. – Igor Belegradek Jan 29 '23 at 21:28
  • @PiotrHajlasz: Soon. – Moishe Kohan Jan 29 '23 at 23:06

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The answer is positive and follows from Corollary 2 in

Palais, Richard S., Extending diffeomorphisms, Proc. Am. Math. Soc. 11, 274-277 (1960). ZBL0095.16502.

(A caveat: Palais is not entirely clear about the degree of smoothness he allows, he just says "differentiable." However, I think, it works for $C^k$-smooth map for every $k>0$.)

Applying Corollary 2 in the case of maps of closed $n$-balls $B^n$ to $S^n$, one obtains that if $\phi, \psi: B^n \to S^n$ are smooth embeddings, then there exists a diffeomorphism $F: S^n\to S^n$ such that $\phi=F\circ \psi$ on $B^n$. Now, take $\psi$ to be the identity embedding $B^n\to S^n$ (where $B^n$ is a hemisphere). Then it follows that $F$ is the desired extension of $\phi$.

PS: For some reason I had Palais' paper open in my browser for a week before you posted the question. :)

Moishe Kohan
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    We (my collaborators and I) inspected Palais' the proof carefully and we wrote our own version of the proof including all details and it works for $C^k$ diffeomorphisms for any $k=1,2,\ldots$. – Piotr Hajlasz Jan 30 '23 at 14:27