Newsgroups: sci.math.research
From: jbaez@BOURBAKI.MIT.EDU (John C. Baez)
Subject: Re: Diffeomorphisms of the 3-ball
Date: Tue, 21 Apr 1992 15:39:09 GMT

>jbaez@BOURBAKI.MIT.EDU (John C. Baez) writes:
>
>>Here's a result I hope is true, and would like to use, but don't know a
>>reference to.  First, every diffeomorphism of S^2 extends to a
>>diffeomorphism of the closed 3-ball, D^3. (?) Second, this extension is
>>unique up to isotopy. (?)  Can anyone help me with these?

Thanks to all for their help.  Here's the straight dope.  Both my
guesses are true.  The first is shown in 

S. Smale, Diffeomorphisms of the 2-sphere, Proc. AMS 10 (1959), 621-626.

and

J. Munkres, Differentiable isotopies on the 2-sphere, Mich. Math. Jour.
7 (1960), 193-197.

As the lengths of these papers indicate, the proofs, while sneaky, are not
apparently not insanely complicated, though I didn't read it.  Smale in
fact shows more, namely that the space of orientation-preserving
diffeomorphisms of S^2 has SO(3) as a strong deformation retract.  The
short way of stating the basic result (that all diffeos of S^2 extend to
the ball) is that the group Gamma_3 vanishes.

For my second guess, Greg Kuperberg assures me that it follows from
Gamma_4 = 0.  For a while this was obvious to me but right now it's not.
Say we know Gamma_4 = 0, i.e. that all diffeos of S^3 extend to D^4.
How do I show that any two diffeos of D^3 that are the identity on S^2
are isotopic?  Well, I can follow my nose with the best of 'em, so I'll
think of S^3 as the union of two D^3's glued along an S^2 in the usual
way (northern and southern hemispheres).  Say I've got two diffeos of
D^3 that are the identity on S^2, f and g.  I'll zap the northern
hemisphere with f and the southern hemisphere with g, getting a diffeo
of S^3.  Why?  So I can use what I've been told, and extend this diffeo
to one of D^4, say F.  Somehow this is supposed to give me diffeos of 
D^3 that "interpolate" between f and g.  I can foliate D^4 with D^3's
that "interpolate" between the northern hemisphere.  But I don't see how
to use this to finish up the job.  Can I finish up the proof without
using any sophisticated tricks, or do you need to be a real topologist
to wrap things up?

In any event, the proof that Gamma_f = 0 fills up a medium-sized book by
J. Cerf, "Sur les diffeomorphisms de la sphere de dimension trois
(Gamma_4 = 0)", Lecture Notes in Math. 53 (1968), Springer-Verlag.

As pointed out, things get a lot weirder in higher dimensions.

==============================================================================

Newsgroups: sci.math.research
From: greg@math.berkeley.edu (Greg Kuperberg)
Subject: Re: Diffeomorphisms of the 3-ball
Date: Sat, 25 Apr 1992 03:03:21 GMT

My first posting on the subject contained a number of errors, both
mathematical and historical.

Firstly, judging by history, triangulating topological 3-manifolds is not
harder than smoothing PL 3-manifolds, because the Hirsch-Smale theorem
(that PL 3-manifolds have a unique smooth structure) antedates the
triangulation theorem (first due to Moise) by eight years.  The "heart" of
the Hirsch-Smale theorem is Smale's theorem that Gamma_3 = 0.

Also, I have confused the notion of a diffeomorphism of the sphere
being isotopic to the identity with the notion of the diffeomorphism
extending to the ball.  In the latter case the diffeomorphism is
pseudo-isotopic to the identity.  If you have two diffeomorphisms
of a manifold M, a pseudo-isotopy is a diffeo of M x I which restricts
to the two diffeos at the boundary.  Whereas an isotopy is a
*level-preserving* diffeo.  (Pseudo-isotopy is Andrew Casson's term;
Geoff Mess calls it a concordance.)

We have a flag of three groups of diffeomorphisms of S^n:

All diffeos    = A_n
    |
Diffeomorphisms pseudo-isotopic to the identity  = B_n
    |
Diffeomorphsims isotopic to the identity  = C_n

So you might say that there is some peace of mind among the intuitive
topologists in any dimension for which A_n = B_n = C_n.
You can make any of the three quotient groups you like, and
Gamma_n-1 is defined as A_n/B_n.  I believe it is
known that B_n = C_n for every n except n=4, where it is open.
However, Andrew Casson tells me that the analogous groups for
other manifolds (perhaps S^3 x S^2 works) are often *not* equal.

Cerf showed directly that A_3 = C_3, a decisively stronger result
than A_3 = B_3.  For n>4, the assertion that A_n = C_n is proved
by rendering them both equal to B_n.

My main semi-interesting comment in this whole affair is that
John Baez's second question concerned isotopy of diffeomorphisms
of a ball B^n, but this is equivalent to considering the above
twoer for S^n.  That is, if A'_n is all diffeos of the ball that
are the identity at the boundary, B'_n is those pseudo-isotopic
to the identity, etc., then in fact A'_n/C'_n = A_n/C_n, and so on.
The proof is similar to the proof of the assertion above that an
extension of a diffeo of S^n to B^n+1 is equivalent to a pseudo-isotopy
of the diffeo of S^n.  (Dealing with the group B_n is a little trickier
but I think it still works.)  Let me prove that assertion to illustrate
the idea:

Suppose you have a diffeo of S^n that extends to a diffeo f of B^(n+1).  
Choose g so that g(f(0)) = 0, the derivative of g(f(x)) at x = 0 is the
identity, and g is the identity at the boundary of the ball.  Now
let h be the map from the complement of the ball to R^n+1 that takes
a vector ru, where u is a unit vector and r is a radius, to the
vector (r-1)u.  Then h^-1(B^n+1) is S^n x [1,2] in spherical coordinates,
and the map h^-1(g(f(h(x))) extends to the boundary and is the desired
pseudo-isotopy.