 
Summary: RESEARCH BLOG 7/20/03
On Friday, Perelman posted a new paper, which claims to finish
off the "elliptization" conjecture, along with the results from [2] and
[1]. He proves that if a closed Riemannian 3manifold M has a prime
decomposition which contains no aspherical factors, then the Ricci flow
with cutoff on M, defined in [1], will become extinct in finite time. If
M has a prime decomposition with no aspherical factors, then M is
a connect sum of copies of S2
× S1
, S2 ~×S1
, and manifolds with finite
fundamental group. It is well known that S2
× S1
and S2 ~×S1
are the
only prime 3manifolds with 2(M) = 0. Also, 1(M) < if and
only if there is a nonzero degree map f : S3
M, so 3(M) = 0
if 1(M) < . Conversely, one may show that if 3(M) = 0, then
either 2(M) = 0, in which case there is an embedded nontrivial S2
