7/31/2019 Completion of the Proof for the Poincar Conjecture

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On the Completion of the Proof for the Poincar Conjecture by C. Y. Lo

It is generally agreed that Hamilton and Perelman make the major contributions to the proof of thePoincare conjecture. However, the completion of the proof was actually done by Cao and Zhu becausethey published a proof for the conjecture in June 2006 [1].

Some objected this conclusion because Cao & Zhu have used an argument that is almost identicalto a result ofKleiner& Lott posted online in 2003. This led to an erratum being issued by Cao & Zhuin the December 2006 issue of the same journal where the original article had appeared. However, thisfact does not change the above conclusion since Cao & Zhu completed the proof first. One might arguethat the result of Kleiner & Lott could 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 resultafter Perelman posted his article on November 11, 2002. From then on, it took Cao & Zhu anotherthree years to get the job done, and to beat Kleiner & Lott, Morgan & Tian 1), etc. 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, thereis no evidence to support such a claim. Perelman's article was called "The Entropy Formula for theRicci Flow and Its Geometric Applications." He did not refer directly to the Poincar conjecture but

rather to Hamilton's concept of the Ricci flow, demonstrating its applicability to the larger Poincarconjecture. Perelman's article was terse and telegraphic, with large gaps in his reasoning, but after hesent 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 Americanuniversities in 2003, Perelman essentially withdrew from public communication, although he wasfriendly enough to reporters intrepid enough to track him down in the labyrinthine streets of his centralSt. 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 LondonGuardian. Thus, Perelman has never completed his proof. In fact, Perelmans article should beconsidered as essentially a collection of sub-conjectures.

If Perelmans article could be considered as a proof, one might as well consider that Poincare hasdone the proof. The difference between Poincare and Perelman is that Perelman made his claim ofhaving 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 Internetpostings because these do not include a valid proof for the conjecture in mathematics. It is more likelythat the internet postings are intended to make the claim first. It is common knowledge in mathematicalanalysis that it is far easier to make a conjecture than provide the actually proof; and one may use thismethod to gain some time to have the credit in a tough competition.

Apparently, Perelman has gained more than four years for this competition. However, Perelmanjust simply lost the race. One may note also the time that Perelman essentially withdrew from publiccommunication is about the time that Kleiner & Lott posted their result on line. Withdrew from publiccommunication would be the best way to hide personal efforts on the competition. Others easily goalong with the story of Perelman because they, particularly the losers, probably do not want to beknown as beaten by Cao & Zhu, in addition to Perelman. In fact, Morgan1) even gave a plenary lecture

at theInternational Congress of MathematiciansinMadrid on August 24, 2006, declaring that "in 2003,Perelman solved the Poincar Conjecture" without presenting any scientific evidence.

Endnotes:

1. In this adventure, the Morgan & Tian team achieves the least among the three teams, but they probably are the mostvocal team. Tian also was a student of S. T. Yau.

Reference:

1. HUAI-DONG CAO & XI-PING ZHU, ASIAN J. MATH. Vol. 10, No. 2, pp. 165492, June 2006.

Summary: Perelmans excellent article is essentially a collection of sub-conjectures of the Poincare conjecture. The completion ofthe proof was actually done by Cao and Zhu because they published a proof for the conjecture in June 2006.

http://en.wikipedia.org/wiki/Bruce_Kleinerhttp://en.wikipedia.org/wiki/Bruce_Kleinerhttp://en.wikipedia.org/wiki/Bruce_Kleinerhttp://en.wikipedia.org/wiki/John_Lott_(mathematician)http://en.wikipedia.org/wiki/International_Congress_of_Mathematicianshttp://en.wikipedia.org/wiki/International_Congress_of_Mathematicianshttp://en.wikipedia.org/wiki/International_Congress_of_Mathematicianshttp://en.wikipedia.org/wiki/Madridhttp://en.wikipedia.org/wiki/Madridhttp://en.wikipedia.org/wiki/Bruce_Kleinerhttp://en.wikipedia.org/wiki/John_Lott_(mathematician)http://en.wikipedia.org/wiki/International_Congress_of_Mathematicianshttp://en.wikipedia.org/wiki/Madrid