[FOM] What's going on with the Poincare conjecture?

[Timothy Y. Chow]

Joe Shipman wrote:

> This month, there has been a flurry of news stories that two Chinese 
> mathematicians, Cao and Zhu, have provided the "final brick" in the 
> proof, and that the Poincare conjecture is now proved. However, there 
> are two disturbing things about this:

For some additional perspective on this matter, one should consult the 
200-page manuscript by Bruce Kleiner and John Lott, "Notes on Perelman's 
papers."  This was posted on the Los Alamos ArXiv shortly before the 
announcement by Cao and Zhu.

  http://arxiv.org/abs/math.DG/0605667

This manuscript is the culmination of years of work by the experts in the 
field, as those who have been tracking Kleiner and Lott's webpage know:

  http://www.math.lsa.umich.edu/~lott/ricciflow/perelman.html

Note that, apart from presenting his papers and his lectures series at MIT 
and Stony Brook, Perelman himself has not been significantly involved in 
these efforts.

Here are a couple of relevant quotes from the Kleiner-Lott paper.

"Pereleman's proofs are concise and, at times, sketchy.  The purpose of 
these notes is to provide the details that are missing in [40] and [41],
which contain Perelman's arguments for the Geometrization Conjecture."

"Regarding the proofs, the papers [40, 41] contain some incorrect 
statements and incomplete arguments, which we have attempted to point out 
to the reader. (Some of the mistakes in [40] were corrected in [41].)  We 
did not find any serious problems, meaning problems that cannot be 
corrected using the methods introduced by Perelman."

The dichotomous question in many people's minds---namely, "Is there a gap, 
or has the proof been checked and found to be correct?"---is never 
directly answered by Kleiner and Lott in so many words; the above quotes 
come closest.

Tim