proof of Geometrization/Poincare Conjecture?

[Zbigniew Fiedorowicz]

A colleague has brought my attention to a preprint recently posted to the Math.
Archive. It is by Grisha Perelman (of the Steklov Institute in St. Petersburg),
entitled "The entropy formula for the Ricci flow and its geometric applications",
which was deposited in the Math. Archive on Nov. 11 and is available via the
link http://front.math.ucdavis.edu/math.DG/0211159.

The introduction to the paper concludes with the following statement:
"Finally in Section 13 we give a brief sketch of the proof of the geometrization
conjecture."

This refers to Thurston's famous Geometrization Conjecture for 3-manifolds,
which in particular implies the Poincare Conjecture. Perelman claims to
carry out Richard Hamilton's program to prove the conjecture using Ricci flow.

Perelman is a well respected differential geometer and is regarded as an
expert on Ricci flow.

Zig Fiedorowicz
Department of Mathematics
The Ohio State University
Columbus, OH 43210

[Greg Kuperberg]

In article <9jsC9.1$t7....@vixen.cso.uiuc.edu>,

Zbigniew Fiedorowicz <fied...@hotmail.com> wrote:
>A colleague has brought my attention to a preprint recently posted
>to the Math. Archive. It is by Grisha Perelman (of the Steklov
>Institute in St. Petersburg), entitled "The entropy formula for the
>Ricci flow and its geometric applications", which was deposited
>in the Math. Archive on Nov. 11 and is available via the link
>http://front.math.ucdavis.edu/math.DG/0211159.
>
>The introduction to the paper concludes with the following statement:
>"Finally in Section 13 we give a brief sketch of the proof of the
>geometrization conjecture."
>
>This refers to Thurston's famous Geometrization Conjecture
>for 3-manifolds, which in particular implies the Poincare
>Conjecture. Perelman claims to carry out Richard Hamilton's program to
>prove the conjecture using Ricci flow.

This is a serious paper with some very interesting results, but the
author does NOT claim to prove the Poincare conjecture. When he says a
"sketch", he clearly explains in the last section that he doesn't know
how to fill in the sketch. He reduces the Geometrization Conjecture to
a series of other conjectures about Ricci flow. And Richard Hamilton
himself may or may not have recognized these conjectures a long time ago.
Regardless it is good to write them down as Perelman has done.

It seems that the Clay Millenium prizes have stoked religious anticipation
of proof of the Poincare conjecture. Every time someone contributes
a paper to the arXiv that mentions the Poincare conjecture, rumors
spread that the Messiah has come. Personally I like the idea of the
Clay prizes, but this is a silly reaction. The right reaction is to
appreciate tangible progress in geometric topology, and not to expect
the Poincare conjecture to be settled miraculously.
--
/\ Greg Kuperberg (UC Davis)
/ \
\ / Visit the Math ArXiv Front at http://front.math.ucdavis.edu/
\/ * All the math that's fit to e-print *

[Zbigniew Fiedorowicz]

Greg Kuperberg wrote in message <6ovC9.76$t7....@vixen.cso.uiuc.edu>:

>This is a serious paper with some very interesting results, but the
>author does NOT claim to prove the Poincare conjecture. When he says a
>"sketch", he clearly explains in the last section that he doesn't know
>how to fill in the sketch. He reduces the Geometrization Conjecture to
>a series of other conjectures about Ricci flow. And Richard Hamilton
>himself may or may not have recognized these conjectures a long time ago.
>Regardless it is good to write them down as Perelman has done.

My understanding of what he is claiming is that, while he can't yet
fully prove Hamilton's conjectures, he can prove enough of them to
get the Geometrization Conjecture.

Earlier in his introduction he writes:

In this paper we carry out some details of Hamilton program. The
more technically complicated arguments, related to the surgery, will
be discussed elsewhere. We have not been able to confirm Hamilton's hope that
the solution that exists for all time $t\to\infty$ necessarily has
bounded normalized curvature; still we are able to show that the
region where this does not hold is locally collapsed with
curvature bounded below; by our earlier (partly unpublished) work
this is enough for topological conclusions.

Moreover in his abstract he writes:
We also verify several assertions related to Richard Hamilton's program
for the proof of Thurston geometrization conjecture for closed three-manifolds,
and give a sketch of an eclectic proof of this conjecture, making use of
earlier results on collapsing with local lower curvature bound.

Perhaps he meant to use some other word than "eclectic", but if you take
him literally, then he is claiming the geometrization conjecture.

Zig Fiedorowicz

[Ian Agol]

Here is Perelman's response to the question of whether he
has proven the claims in his paper which imply the geometric
decomposition conjecture:

1. is correct, 2. is correct with n replaced by 3/2 (the statement has to
be scale invariant, and n occured here by misprint), and only for maximal
horns.
Grisha

On Tue, 19 Nov 2002, Ian Agol wrote:

> Hi,
> I wanted to double check with you
> that you are claiming to have a proof of
> the geometrization conjecture. Specifically,
> you are claiming to have a proof that
>
> 1. You may do surgeries on horns so that
> there are only finitely many in a given
> time interval
>
> 2. The volumes of horns are > cT^n (for
> general n?) and this shows that the Ricci
> flow will be non-singular after finitely
> many of these surgeries.
>
> thanks,
> Ian Agol