Describing some of Henri Poincare's work, author Donal O'Shea writes on page 132 that Poincare "produced an infinite family of closed three-dimensional manifolds that are not homeomorphic to one another and showed that one can have nonhomeomorphic manifolds with the same Betti numbers and, in fact, with the same Betti numbers as a sphere." Oh, so you don't know what a Betti number is? Well, O'Shea's Glossary of Terms describes it as "an integer counting that number of inequivalent manifolds of a given dimension in a manifold that do not bound a submanifold of one dimension higher." If this is not quite your cup of mathematical tea (as it is not quite mine, despite my B.S. in mathematics from a highly-regarded engineering school), then THE POINCARE CONJECTURE might just be a full teapot that you want to skip.
A fundamental rule of nonfiction is to identify and write for your intended audience, and it is difficult to imagine who O'Shea saw here as his audience. THE POINCARE CONJECTURE addresses a mathematically famous but publicly obscure hypothesis from the general field of topology, the study of shapes and curvature. Filled to overflowing with historical background, dating back to Euclid's original five postulates for plane geometry, the main body of O'Shea's book is exclusively textual. Even the hypercritical Ricci flow equation, the main vehicle through which Grigory Perelman achieved his landmark proof, is relegated in abbreviated form to the footnotes. All of this, combined with O'Shea's opening chapters attempting to explain topological manifolds, suggests a book targeted at nonmathematicians. Yet as the excerpt above demonstrates, the author seems too often not to have found the necessary nonmathematical explication to reach successfully a nonprofessional mathematical audience. Conversely, those who are sufficiently mathematically versed to follow O'Shea's elucidation will likely find it far too light in pure mathematical content.
In point of fact, author O'Shea defers even stating or describing the Poincare Conjecture until page 45: "...the assertion that any compact three-manifold on which any closed path can be shrunk to a point, is the same topologically as (that is, homeomorphic to) the three-sphere." What follows is 136 pages of interesting mathematical history tracing from Euclid through Gauss, Bolyai, and Lobachevsky to Riemann, Felix Klein, Poincare himself, and then on to David Hilbert, Smale, Thurston, Richard Hamilton, and finally, Perelman. The final eighteen pages attempt in some small way to capture the excitement behind Perelman's revelation of the solution to one of the Clay Institute's seven millennium problems (each with an accompanying million dollar prize), but the conclusion is disappointingly anticlimactic. Perelman being the professional hermit and cipher to which he apparently aspires, O'Shea gives us little sense of the man and his background and virtually no sense of his labors (and, presumably, travails) over this infamous hypothesis. Perelman appears in this book as he appeared before the mathematical community in 2003 - on the stage for too precious few moments and then whisked away, backed to self-selected obscurity in Russia. Compare this to Simon Singh's brilliant treatment of Andrew Wiles and his search for the proof of the so-called Fermat's Last Theorem.
In that same final eighteen pages, O'Shea returns to the "real world" question that his book's subtitle suggests would be answered as a result of proving Poincare's Conjecture: the shape of the universe. He devotes slightly over one page to revealing his answer -- that "the question...is still very much open." The teaser subtitle is just that, a teaser for the scientifically interested and inclined. Gee, thanks.
THE POINCARE CONJECTURE is long on the history of non-Euclidian geometry and the various subdisciplines of topology and unsatisfying short on the actual proof of its eponymous theorem, the peculiar man behind that proof, and the real world implications of the result. The text itself runs 200 pages but offers just short of 100 more space-filling pages larded with footnotes, glossary of terms, glossary of names, timeline, bibliography, further reading, art credits, acknowledgments, and an index. Frankly, the New Yorker's 14-page article about Perelman in August 2006 by Nasar and Gruber was more informative. For this book, a gentleman's 3 stars is the most I can muster.