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Is the Poincaré Conjecture Proved At Last?

*By Fernando Q. Gouvêa*

The Poincaré Conjecture, one of the Clay Mathematics Institute's million-dollar Millennium Problems, may be settled at last. In 2002 and 2003, Grigori Perelman of the Steklov Institute in St. Petersburg posted several papers to an online preprint archive in which he claimed to have proved Thurston's Geometrization Conjecture, which is known to imply Poincaré. Specialists have been reading his papers and developing his ideas since then. Two recent papers, published almost simultaneously, have claimed a complete proof based on Perelman's ideas. One is "Notes on Perelman's Papers," by Bruce Kleiner and John Lott (posted to the ArXiv preprint server on May 25, 2006); the other is "A Complete Proof of the Poincaré and Geometrization Conjectures-Application of the Hamilton-Perelman Theory of the Ricci Flow," by Huai-Dong Cao and Xi-Ping Zhu (published in the June 2006 issue of the *Asian Journal of Mathematics).*

To understand the conjecture, start with the three-sphere S^{3}, i.e., the set of points (x1, x2, x3, x4) in R^{4}satisfying the equation x_{1}² x_{2}² x_{3}² x_{4}² = 1. This is perhaps the simplest compact three-manifold. One way in which it is simple is the fact that any loop embedded in it can be continuously deformed to a point. Poincaré conjectured, early in the 20th century, that this property in fact characterizes the three-sphere, that is, that any other compact three-dimensional manifold with the loop contracting property would have to be homeomorphic to S³. While very plausible, this conjecture turned out to be extraordinarily difficult to prove.

Analogues of the conjecture in dimension five and higher were proved in the 1960s by Smale, Stallings, and Wallace. The four-dimensional version took twenty more years, being proved by Freedman in the 1980s. It now seems that the original three-dimensional conjecture has finally been settled. Perelman's methods are based on ideas of Richard Hamilton on the "Ricci flow" and on William Thurston's high-powered "Geometrization Conjecture," which provides a very general description of all closed three-manifolds with finite fundamental group. Thurston's conjecture contains the Poincaré Conjecture as a special case. It is the Geometrization Conjecture that seems to have been proved, therefore settling also the Poincaré Conjecture.

Interest in the conjecture was heightened by the fact that in 2000 the Clay Mathematics Institute made it one of its seven Millennium Problems, offering one million dollars for its solution. One can read more about the conjecture and the Millennium Problems at the CMI web site at http://www.claymath.org/millennium/Poincare_Conjecture/.

Curiously, it is unclear whether Perelman will win the prize. The rules for the Millennium Prizes specify that the work must be published in a refereed journal and must survive the examination of the mathematical community for at least a two-year period. So far, Perelman shows no indication of any intention to publish his papers. Perelman's preprints are fairly sketchy and would probably not be publishable as they stand. The two exegeses by Kleiner and Lott and by Cao and Zhu are immensely longer than the Perelman's original papers, and both claim to contain the full details of the proof, leaving an interesting knot for the Clay Institute to disentangle.

In addition to the CMI site noted above, one should check the site maintained by Kleiner and Lott, http://www.math.lsa.umich.edu/~lott/ricciflow/perelman.html , which contains links to many other online materials related to the work of Perelman. See also the news story published in the *Wall Street Journal* on July 21, 2006 (page A9). Details on all the Millennium Problems can be found in Keith Devlin's *The Millennium Problems* and in the recent official publication from the American Mathematical Society and the Clay Institute, The Millennium Prize Problems, ed. by J. Carlson, A. Jaffe and A. Wiles. Devlin's account of the conjecture is aimed at the general public, while the article by Milnor in the official volume is more technical (but still quite readable). *The Asian Journal of Mathematics* is at http://www.ims.cuhk.edu.hk/~ajm/ and the ArXiv preprint server is http://arxiv.org.