 
Summary: RESEARCH BLOG 10/8/03
Perelman states in his second paper that one of his goals is to make
a canonical Ricci flow, which exists on a maximal subset of space time.
One possible application of this would be to the Smale conjecture (and
its generalizations), although it should also follow if one can perform
the surgeries continuously. For the S3
, the Smale conjecture states
that the space of diffeomorphisms is homotopy equivalent to O(4), the
group of isometries of the round 3sphere. This was proven by Hatcher
[2]. Similarly, for a hyperbolic manifold, the group of diffeomorphisms
is homotopy equivalent to the isometry group. This was proven by
Gabai [1]. There are similar statements for other manifolds satisfying
the geometrization conjecture, most of which have been solved. To
try to prove this using Ricci flow, one could use Gabai's strategy. It
is equivalent to show that all the homotopy groups of Diff(M) are
trivial, since it is known that Diff(M) is homotopy equivalent to a cell
complex. Gabai starts with a hyperbolic manifold (M, g), and considers
an Sn
family of diffeomorphisms, f : Sn
Diff(M). Then f
