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

Timothy Y. Chow tchow at alum.mit.edu
Mon Jun 19 09:50:38 EDT 2006

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.


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:


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.


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