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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

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    $\begingroup$ 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", ams.org/journals/proc/1960-011-02/S0002-9939-1960-0117741-0/…. $\endgroup$
    – Igor Belegradek
    Commented Jan 29, 2023 at 19:19
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    $\begingroup$ 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. $\endgroup$
    – Moishe Kohan
    Commented Jan 29, 2023 at 19:31
  • $\begingroup$ @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? $\endgroup$
    – Igor Belegradek
    Commented Jan 29, 2023 at 20:04
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    $\begingroup$ @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. $\endgroup$
    – Piotr Hajlasz
    Commented Jan 29, 2023 at 20:08
  • $\begingroup$ 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. $\endgroup$
    – Igor Belegradek
    Commented Jan 29, 2023 at 20:23

1 Answer 1

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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. :)

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    $\begingroup$ 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 $k=1,2,\ldots$. $\endgroup$
    – Piotr Hajlasz
    Commented Jan 30, 2023 at 14:27

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