On the Completion of Proof for the Poincaré Conjecture
It is generally agreed that Hamilton and Perelman make the major contributions to the proof of the Poincare conjecture. However, the completion of the proof was actually done by Cao and Zhu because they published a proof for the conjecture in June 2006.
Some objected this conclusion because Cao & Zhu have used an argument that is almost identical to a result of Kleiner & Lott posted online in 2003. This led to an erratum being issued by Cao & Zhu in the December 2006 issue of the same journal where the original article had appeared. However, this fact does not change the above conclusion since Cao & Zhu completed the proof first. One might argue that the result of Kleiner & Lott be crucial, and thus they should have more credit to the proof. Such an argument can hardly stand since it took Kleiner & Lott only about a year to obtain their result after Perelman posted his article on November 11, 2002. From then on, it took Cao & Zhu another three years to get the job done, and to beat Kleiner & Lott in this competition. Thus, this should have made clear that Cao & Zhu have done more hard work than Kleiner & Lott.
Some claimed that Perelman had completed the proof for the Poincare conjecture. However, there is no evidence to support such a claim. Perelman's article was called "The Entropy Formula for the Ricci Flow and Its Geometric Applications." He did not refer directly to the Poincaré conjecture but rather to Hamilton's concept of the Ricci flow, demonstrating its applicability to the larger Poincaré conjecture. Perelman's article was terse and telegraphic, with large gaps in his reasoning, but after he sent e-mails to a few of his former colleagues they sensed the importance of his discovery. However, Perelman offered nothing other than his three Internet postings. After a series of lectures at American universities in 2003, Perelman essentially withdrew from public communication, although he was friendly enough to reporters intrepid enough to track him down in the labyrinthine streets of his central St. Petersburg neighborhood. "He placed the papers on the web archive and basically said 'that's it,'" Oxford University mathematics professor Nigel Hitchin told James Randerson of the London Guardian. Thus, Perelman has never completed his proof. In fact, Perelman’s article should be considered as essentially a collection of sub-conjectures.
If Perelman’s article could be considered as a proof, one might as well consider that Poincare has done the proof. The difference between Poincare and Perelman is that Perelman made his claim of having completed the proof whereas Poincare did not. However, Perelman has a problem in his credibility on this because he (or anybody) still has not completed one of his sub-conjectures.
It was difficult to believe that Perelman had intended to do no more than beyond his three Internet postings because these do not include a valid proof for the conjecture in mathematics. It is more likely that the internet postings are intended to make the claim first. It is common knowledge in mathematical analysis that it is far easier to make a conjecture than provide the actually proof; and one may use this method to gain some time to have the credit in a tough competition.
Apparently, Perelman has gained more than four years for this competition. However, Perelman just simply lost the race. One may note also the time that Perelman essentially withdrew from public communication is about the time that Kleiner & Lott posted their result on line. Withdrew from public communication would be the best way to hide personal efforts on the competition. Others easily go along with the story of Perelman because they probably do not want to be known as beaten by others, in addition to Perelman.
