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3 4th question added; typo corrected; deleted 8 characters in body

A fundamental result in 3-dimensional smooth topology, which in computer jargon we might refer to as "a primitive", is the statement that any ($C^\infty$) diffeomorphism of S2 extends to a diffeomorphism of the closed 3-ball D3. Another way of stating the same thing would be to say that $Diff(S^2)$ is a connectedspace. This theorem was first proven by Munkres [Mich. Math. Jour. 7 (1960), 193-197]. Later, Smale proved the stronger result that $Diff(S^2)$ has the homotopy type of $O(3)$ [Proc. AMS 10 (1959), 621-626]. Another proof of Smale's result is given by Cerf in the appendix to [Sur les difféomorphismes de la sphère de dimension trois ($Γ_4=0$), Lecture Notes in Mathematics, No. 53. Springer-Verlag, Berlin-New York 1968].

Question 1: Are there any other known proofs to the statement that any diffeomorphism of S2 extends to a diffeomorphism of the closed 3-ball D3?

There are two reasons I'm not fully happy with the proofs I cited above. Smale's proof and Cerf's proof show much more and use what looks to me like "too much machinery" for just the "$Diff(S^2)$ is contractibleconnected" statement, and, in particular, machinery which seems outside basic differential topology (maybe I'm wrong; I haven't gone into them in much detail). Munkres's proof has a number of back-references to another of his papers [Ann. Math. 72(3) (1960), 521-554], and corners need to be smoothed over and over and over and over again to get an honest smooth isotopy between a given diffeomorphism of S2 and the identity. What is worse, it seems difficult to extract an algorithm from Munkres's proof (Lemma 1.1 looks non-constructive- I wouldn't know how to extract a concrete diffeomorphism out of its proof), which brings me to my second question:

Question 2: How could I implement an extension of a smooth diffeomorphism of the 2-sphere to the 3-disc? To make things really concrete, let's say I had an image of the surface of the earth which I deformed by some strange diffeomorphism f of S2. How (by computer) could I smoothly deform it back to the usual picture of the earth?

One dimension down, maybe one way to do it might be to "relax a diffeomorphism of a circle gradually using the heat equation" (see Greg Kuperberg's comment here). Does this work one dimension up? I couldn't figure this out, but I don't see an obvious obstruction- not in dimension $3$. Or maybe there's a slick way of implementing Munkres's proof by lifting an orientation-preserving diffeomorphism of $S^2$ to $Spin(3)$ or something... I really have no idea.

Note, though, that other proofs that diffeomorphisms of S1 extend to D2 clearly seem to fail in dimension 3... in particular, trying to use some sort of Alexander trick to comb all the "bad parts" of the diffeomorphism into a small disc and shrink that disc to a point will not give rise to a smooth isotopy.

Finally, Morris Hirsch says in a footnote on Page 38 of The Collected Papers of Stephen Smale: "Around this time [1959] an outline of a proof attributed to Kneser was circulating by word of mouth; it was based on an alleged version of the Riemann Mapping Theorem which gives smoothness at the boundary of smooth Jordan domains, and smooth dependence on parameters. I do not know if such a proof was ever published."

Question 3: Was such a proof ever published? Is there anything else to be said about this proof outline?

Edit: Actually, I'd like to add even a fourth question:

Question 4: Are there any "second generation" detailed expositions of any of the above proofs?

A fundamental result in 3-dimensional smooth topology, which in computer jargon we might refer to as "a primitive", is the statement that any ($C^\infty$) diffeomorphism of S2 extends to a diffeomorphism of the closed 3-ball D3. Another way of stating the same thing would be to say that $Diff(S^2)$ is a connected space. This theorem was first proven by Munkres [Mich. Math. Jour. 7 (1960), 193-197]. Later, Smale proved the stronger result that $Diff(S^2)$ has the homotopy type of $O(3)$ [Proc. AMS 10 (1959), 621-626]. Another proof of Smale's result is given by Cerf in the appendix to [Sur les difféomorphismes de la sphère de dimension trois ($Γ_4=0$), Lecture Notes in Mathematics, No. 53. Springer-Verlag, Berlin-New York 1968].

Question 1: Are there any other known proofs to the statement that any diffeomorphism of S2 extends to a diffeomorphism of the closed 3-ball D3?

There are two reasons I'm not fully happy with the proofs I cited above. Smale's proof and Cerf's proof show much more and use what looks to me like "too much machinery" for just the "$Diff(S^2)$ is contractible" statement, and, in particular, machinery which seems outside basic differential topology (maybe I'm wrong; I haven't gone into them in much detail). Munkres's proof has a number of back-references to another of his papers [Ann. Math. 72(3) (1960), 521-554], and corners need to be smoothed over and over and over and over again to get an honest smooth isotopy between a given diffeomorphism of S2 and the identity. What is worse, it seems difficult to extract an algorithm from Munkres's proof (Lemma 1.1 looks non-constructive- I wouldn't know how to extract a concrete diffeomorphism out of its proof), which brings me to my second question:

Question 2: How could I implement an extension of a smooth diffeomorphism of the 2-sphere to the 3-disc? To make things really concrete, let's say I had an image of the surface of the earth which I deformed by some strange diffeomorphism f of S2. How (by computer) could I smoothly deform it back to the usual picture of the earth?

One dimension down, maybe one way to do it might be to "relax a diffeomorphism of a circle gradually using the heat equation" (see Greg Kuperberg's comment here). Does this work one dimension up? I couldn't figure this out, but I don't see an obvious obstruction- not in dimension $3$. Or maybe there's a slick way of implementing Munkres's proof by lifting an orientation-preserving diffeomorphism of $S^2$ to $Spin(3)$ or something... I really have no idea.

Note, though, that other proofs that diffeomorphisms of S1 extend to D2 clearly seem to fail in dimension 3... in particular, trying to use some sort of Alexander trick to comb all the "bad parts" of the diffeomorphism into a small disc and shrink that disc to a point will not give rise to a smooth isotopy.

Finally, Morris Hirsch says in a footnote on Page 38 of The Collected Papers of Stephen Smale: "Around this time [1959] an outline of a proof attributed to Kneser was circulating by word of mouth; it was based on an alleged version of the Riemann Mapping Theorem which gives smoothness at the boundary of smooth Jordan domains, and smooth dependence on parameters. I do not know if such a proof was ever published."

Question 3: Was such a proof ever published? Is there anything else to be said about this proof outline?

Edit: Actually, I'd like to add even a fourth question:

Question 4: Are there any "second generation" detailed expositions of any of the above proofs?
1

# Extending a diffeomorphism of S^2 to D^3

A fundamental result in 3-dimensional smooth topology, which in computer jargon we might refer to as "a primitive", is the statement that any ($C^\infty$) diffeomorphism of S2 extends to a diffeomorphism of the closed 3-ball D3. Another way of stating the same thing would be to say that $Diff(S^2)$ is a connected space. This theorem was first proven by Munkres [Mich. Math. Jour. 7 (1960), 193-197]. Later, Smale proved the stronger result that $Diff(S^2)$ has the homotopy type of $O(3)$ [Proc. AMS 10 (1959), 621-626]. Another proof of Smale's result is given by Cerf in the appendix to [Sur les difféomorphismes de la sphère de dimension trois ($Γ_4=0$), Lecture Notes in Mathematics, No. 53. Springer-Verlag, Berlin-New York 1968].

Question 1: Are there any other known proofs to the statement that any diffeomorphism of S2 extends to a diffeomorphism of the closed 3-ball D3?

There are two reasons I'm not fully happy with the proofs I cited above. Smale's proof and Cerf's proof show much more and use what looks to me like "too much machinery" for just the "$Diff(S^2)$ is contractible" statement, and, in particular, machinery which seems outside basic differential topology (maybe I'm wrong; I haven't gone into them in much detail). Munkres's proof has a number of back-references to another of his papers [Ann. Math. 72(3) (1960), 521-554], and corners need to be smoothed over and over and over and over again to get an honest smooth isotopy between a given diffeomorphism of S2 and the identity. What is worse, it seems difficult to extract an algorithm from Munkres's proof (Lemma 1.1 looks non-constructive- I wouldn't know how to extract a concrete diffeomorphism out of its proof), which brings me to my second question:

Question 2: How could I implement an extension of a smooth diffeomorphism of the 2-sphere to the 3-disc? To make things really concrete, let's say I had an image of the surface of the earth which I deformed by some strange diffeomorphism f of S2. How (by computer) could I smoothly deform it back to the usual picture of the earth?

One dimension down, maybe one way to do it might be to "relax a diffeomorphism of a circle gradually using the heat equation" (see Greg Kuperberg's comment here). Does this work one dimension up? I couldn't figure this out, but I don't see an obvious obstruction- not in dimension $3$. Or maybe there's a slick way of implementing Munkres's proof by lifting an orientation-preserving diffeomorphism of $S^2$ to $Spin(3)$ or something... I really have no idea.

Note, though, that other proofs that diffeomorphisms of S1 extend to D2 clearly seem to fail in dimension 3... in particular, trying to use some sort of Alexander trick to comb all the "bad parts" of the diffeomorphism into a small disc and shrink that disc to a point will not give rise to a smooth isotopy.

Finally, Morris Hirsch says in a footnote on Page 38 of The Collected Papers of Stephen Smale: "Around this time [1959] an outline of a proof attributed to Kneser was circulating by word of mouth; it was based on an alleged version of the Riemann Mapping Theorem which gives smoothness at the boundary of smooth Jordan domains, and smooth dependence on parameters. I do not know if such a proof was ever published."

Question 3: Was such a proof ever published? Is there anything else to be said about this proof outline?