# Extending diffeomorphisms

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

• 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/…. Jan 29 at 19:19
• 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. Jan 29 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? Jan 29 at 20:04
• @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. Jan 29 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. Jan 29 at 20:23

(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 $$\phi$$.
• 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 $k=1,2,\ldots$. Jan 30 at 14:27