Mapping Class Group computations
Blake Winter
MCG(R^n)={1}?
Also, is MCG(S^4)={1}? What about MCG(S^n), n>1?
bury....@gmail.com
Your questions in order:
1) That the group of diffeomorphisms of R^n has the homotopy-type of
O(n) has been known going back all the way to Cerf's dissertation,
perhaps even earlier. The standard reduction goes like this:
Diff(R^n) deformation-retracts to the subspaces Diff(R^n,0) ie:
diffeomorphisms that fix 0. Diff(R^n,0) has as a deformation-retract
the subspace of linear automorphisms of R^n. GL_n has as a
deformation-retract O_n. The 1st step is pretty standard (translation
homotopy). The 2nd step is to consider the homotopy: (x,t) --> f(tx)/
t (for t in (0,1]) and define it to be f'(0)(x) for t=0 ie: the
derivative of f at 0 in the direction x. The 3rd step is Gram-Schmidt.
I believe the relevant bit of Cerf's dissertation appears in
"Topologie de certains espaces de plongements," Bull SMF tome 89
(1961). This observation is vital to the homotopy-classification of
spaces of tubular neighbourhoods of a submanifold of a manifold.
2) MCG(S^4)={1} is an open problem. So if you find a rigorous proof
of this you should publish it.
3) MCG(S^n) for n>4 is closely related to the group of homotopy (n+1)-
spheres. The h-cobordism theorem implies the correspondence. See
Kosinski's textbook, or Milnor's notes on the h-cobordism theorem. So
among other things, these groups tend to be fairly non-trivial.
MCG(S^3)={1} was proven by Cerf. MCG(S^2)={1} I'm not sure who first
proved this, I would guess Schoenflies or somebody of that era. Smale
first proved that Diff(S^2) has the homotopy-type of O(3).
MCG(S^1)={1} is a fun exercise to give students in an intro manifolds
course, similarly proving Diff(S^1) has the homotopy-type of O(2).
-rb