Book Review The Poincaré Conjecture and Poincaré’s Prize Reviewed by W. B. Raymond Lickorish can be shrunk to a point. Armed with a gradu- The Poincaré Conjecture: In Search of the Shape ate course in topology, and perhaps also aware of the Universe that any three-manifold can both be triangulated Donal O’Shea Walker, March 2007 and given a differential structure, an adventurer 304 pages, US$15.95, ISBN 978-0802706545 could set out to find fame and fortune by con- quering the Poincaré Conjecture. Gradually the Poincaré’s Prize: The Hundred-Year Quest to conjecture acquired a draconian reputation: many Solve One of Math’s Greatest Puzzles were its victims. Early attempts at the conjec- George G. Szpiro ture produced some interesting incidental results: Dutton, June 2007 Poincaré himself, when grappling with the proper 320 pages, US$24.95, ISBN 978-0525950240 statement of the problem, discovered his dodec- ahedral homology three-sphere that is not S3; J. H. C. Whitehead, some thirty years later, dis- Here are two books that, placing their narratives carded his own erroneous solution on discovering in abundant historical context and avoiding all a contractible (all homotopy groups trivial) three- symbols and formulae, tell of the triumph of Grig- manifold not homeomorphic to R3. However, for ory Perelman. He, by developing ideas of Richard the last half-century, the conjecture has too often Hamilton concerning curvature, has given an affir- enticed devotees into a fruitless addiction with mative solution to the famous problem known as regrettable consequences. the Poincaré Conjecture. This conjecture, posed The allure of the Poincaré Conjecture has cer- as a question by Henri Poincaré in 1904, was tainly provided an underlying motivation for much a fundamental question about three-dimensional research in three-manifolds. Nevertheless, three- topology. It proposed that any closed, simply con- manifold theory long ago largely bypassed the nected, three-dimensional manifold M be home- conjecture. This resulted from a theorem that omorphic to S3, the standard three-dimensional H. Kneser proved in 1928. It asserts that any com- sphere. The conjecture is therefore fairly easy to pact three-manifold, other than S3, can be cut into state and yet, unlike various other conjectures pieces along a finite collection of two-dimensional in, say, number theory, just a little mathematical spheres, so that, if those spheres are capped off by knowledge is needed to understand the concepts gluing three-dimensional balls to them, the result- involved. The given conditions on M are that it ing pieces are prime. A connected three-manifold is a Hausdorff topological space, each point hav- is prime if it is not S3 and if, whenever a sphere ing a neighbourhood homeomorphic to R3, it is embedded in it separates it into two parts, one of compact and path connected, and any loop in M those parts is a ball. Later J. W. Milnor proved an orientable manifold determines its prime pieces, W. B. Raymond Lickorish is Emeritus Professor of Geomet- its summands, uniquely. Thus it is reasonable to ric Topology at the University of Cambridge. His email restrict study to prime manifolds. A prime three- address is [email protected]. manifold might have been a counterexample to January 2008 Notices of the AMS 37 the Poincaré Conjecture but If the singularities can be shown to correspond to otherwise, having no prop- the spheres and tori that decompose the manifold, er summands at all, it could as already described, cuts can be made and the not have contained within process restarted on the resulting pieces. itself (as a summand) a pos- The book by Szpiro is a fast-moving account sible counterexample to the of mathematics in the making. He presents the conjecture. story of the Poincaré Conjecture, from its formu- Both of these books lation to its solution, in terms of the people who, have to explain, to non- for good or ill, became involved with it. Their mathematicians, the meaning lives are painted with all available colour, with of the statement of the details of childhood and education, with accounts Poincaré Conjecture and of their early mathematical achievements, their something about curvature in hopes, their frustrations, and their disappoint- order to introduce the idea of ments. Mathematicians are shown to have very the Hamilton-Perelman tech- human passions and fears, they too are caught nique. However, Perelman’s up with the conventions, the pressures, and the work also encompasses a politics of their times, and are involved in the proof of the Thurston Ge- tragedies of war and repression. Details marginal- ometrisation Conjecture, a much more significant ly relevant to the Poincaré Conjecture are included achievement, and explaining the meaning of this if they might illuminate personalities involved. is a much more difficult task. A modification Szpiro has done a great job of biographical re- of Kneser’s argument allows a prime three- search and produced an account that will fill many manifold to be cut, by a finite collection of a gap in a mathematician’s historical knowledge. (incompressible) tori, into pieces that contain For many it will be an irresistible opportunity no more such tori (other than parallel copies to relive contentious times past. The drama of of tori of the finite set). After a small manoeu- the announcement of Fields Medal winners at the vre the collection of tori is unique; that is the International Congress in Madrid, the refusal by Jaco-Shalen-Johannson theorem. The conjecture of Perelman to accept a medal or participate in the W. P. Thurston is that this set of tori cuts the prime congress, the uncertainty over a possible award manifold into pieces each of which has one of eight of a one-million dollar Millennium Problem prize possible geometric structures. The Poincaré Con- by the Clay Institute, all these happenings give a jecture is a special case of this: in a simply breathtaking climax to the century-long saga. It connected manifold the set of tori is empty, and seems doubtful if the reclusive Perelman would so the whole manifold should have a geometric want to star in this academic soap opera but a structure which could only be the spherical one. journalist must, it seems, report what he learns. Thurston proved his conjecture for sufficiently Any educated person with a little curiosity will large (or Haken) manifolds, a class for which be able to enjoy this book. He or she will not F. Waldhausen had already shown that homotopy become bogged down by obtrusive mathematical equivalence implies homeomorphism. The eight detail, but will be swept along by this tale of the standard geometries are well-known (and well- all-consuming desire to know the truth and the documented by G. P. Scott). The conjecture is then competition to achieve fame thereby, only to find that each piece of the manifold is the quotient of applause eschewed by the eventual winner. Here one of the standard geometries by a discrete group is an insight into how scientific research really of isometries (acting freely and properly discontin- happens and what an emotional affair it can be; uously), the quotient being complete. Most of the mathematics does need books such as this. mystery of this has centred around understand- The book will be a valuable historical account, ing hyperbolic geometry; pieces with the other a work of reference on the lives of a most im- geometries have Seifert-fibred structures that are pressive number of early topologists. The quantity understood. In very broad outline, the Hamilton and spread of the material covered is ambitious approach, brought to fruition by Perelman, starts and truly impressive. After a brief nod towards by giving a closed three-manifold a smooth struc- Christopher Columbus and the flat earth hypoth- ture with arbitrary Riemannian metric (that is esis, there follows a 35-page account of the life easy). Then the metric is allowed to change in of Henri Poincaré (1854–1912). His childhood, his time according to a Ricci flow equation that is distinguished relatives, his schooling and experi- an analogue of the heat equation of physics. If ence of the Franco–Prussian war, his entry to the all goes very well, the metric will converge to a École Polytechnique in Paris and his three years at metric of constant curvature (just as a heated body the École des Mines are all described. There is a eventually has uniform temperature) and then the long account of his short experience in the Service manifold has one of the eight geometries. Other- des Mines (miner’s lamp number 476 caused the wise it should tend to a metric-with-singularities. pit explosion). Finally he becomes a professor at 38 Notices of the AMS Volume 55, Number 1 the Sorbonne. Next is the story of his winning the manifold …to skeletons, em- prize offered by King Oscar II of Sweden for an bedding the skeletons in two essay on the n-body problem, of his dealings on balls, separating the skin this with G. Mittag-Leffler and of the discovery from the inside of one of of an error in his essay. The famous paper on the balls, then recombining Analysis situs and its complements are given due some of the items. And, prominence, as is the discovery by P. Heegaard of since a ball is a ball is a a mistake in it. There is much here about Hee- ball …”. This is nonsense, gaard but no mention of the Heegaard splitting and yet it is indeed strangely of a three-manifold (still much studied in three- reminiscent of Stallings’ work manifold theory and the basis for the topical and on the high-dimensional gen- very productive Heegaard-Floer homology); this is eralisation of the Poincaré surely the chief thing for which he is remembered Conjecture (atopic, hard for a today.