A fundamental result in three-dimensional smooth topology, which in computer jargon we might refer to as "a primitive", is the statement that any ($C^\infty$) diffeomorphism of $S^2$ extends to a diffeomorphism of the closed three-ball $D^3$. Equivalently: $\mathrm{Diff}(S^2)$ is connected. This theorem was first proven by Munkres [Mich. Math. Jour. 7 (1960), 193-197]. Later, Smale proved the stronger result that $\mathrm{Diff}(S^2)$ 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 of the statement that any diffeomorphism of the two-sphere $S^2$ extends to a diffeomorphism of the closed three-ball $D^3$?

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 "$\mathrm{Diff}(S^2)$ is connected" 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 $S^2$ 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 two-sphere to the three-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 $S^2$. 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 three. Or maybe there's a slick way of implementing Munkres's proof by lifting an orientation-preserving diffeomorphism of $S^2$ to $\mathrm{Spin}(3)$ or something... I really have no idea.

Note, though, that other proofs that diffeomorphisms of $S^1$ extend to $D^2$ clearly seem to fail in dimension three... 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 three-dimensional smooth topology, which in computer jargon we might refer to as "a primitive", is the statement that any ($C^\infty$) diffeomorphism of the two-sphere $S^2$ extends to a diffeomorphism of the closed three-ball $D^3$. Equivalently: $\mathrm{Diff}(S^2)$ is connected. This theorem was first proven by Munkres [Mich. Math. Jour. 7 (1960), 193-197]. Later, Smale proved the stronger result that $\mathrm{Diff}(S^2)$ 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 of the statement that any diffeomorphism of the two-sphere $S^2$ extends to a diffeomorphism of the closed three-ball $D^3$?

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 "$\mathrm{Diff}(S^2)$ is connected" 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 $S^2$ 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 two-sphere to the three-ball? 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 $S^2$. 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 three. Or maybe there's a slick way of implementing Munkres's proof by lifting an orientation-preserving diffeomorphism of $S^2$ to $\mathrm{Spin}(3)$ or something... I really have no idea.

Note, though, that other proofs that diffeomorphisms of $S^1$ extend to $D^2$ clearly seem to fail in dimension three... 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 connected. 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 connected" 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.

A fundamental result in three-dimensional smooth topology, which in computer jargon we might refer to as "a primitive", is the statement that any ($C^\infty$) diffeomorphism of $S^2$ extends to a diffeomorphism of the closed three-ball $D^3$. Equivalently: $\mathrm{Diff}(S^2)$ is connected. This theorem was first proven by Munkres [Mich. Math. Jour. 7 (1960), 193-197]. Later, Smale proved the stronger result that $\mathrm{Diff}(S^2)$ 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 of the statement that any diffeomorphism of the two-sphere $S^2$ extends to a diffeomorphism of the closed three-ball $D^3$?

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 "$\mathrm{Diff}(S^2)$ is connected" 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 $S^2$ 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 two-sphere to the three-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 $S^2$. 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 three. Or maybe there's a slick way of implementing Munkres's proof by lifting an orientation-preserving diffeomorphism of $S^2$ to $\mathrm{Spin}(3)$ or something... I really have no idea.

Note, though, that other proofs that diffeomorphisms of $S^1$ extend to $D^2$ clearly seem to fail in dimension three... 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.

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].

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:

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 connected. 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].

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 connected" 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: