It is a well-known fact that our universe has three dimensions of space. Imagine for a moment that it had only two, like E.A. Abbott’s Flatland, or A.K. Dewdney’s Planiverse. What shape might it have? Well, it might be flat, like an infinite sheet of paper on an infinite tabletop; or it might curve round on itself like the surface of a sphere, with a definite, finite area. It might also be any one of a menagerie of other shapes that mathematicians can tell you about, some finite in area and some infinite.
As two-dimensional creatures stuck in that universe, unable to view it from the outside, how could we deduce its shape? The first step would be to develop a good, robust mathematical theory of two-dimensional spaces in general, encompassing the flat plane, the sphere-surface, and all the others. The same applies to our three-dimensional universe. To find out its shape, we first need a general, abstract mathematical theory of three-dimensional spaces.
That theory is the topic of Donal O’Shea’s new book, The Poincaré Conjecture. The word “space” actually belongs to physics. The abstract mathematical object used to model a space is called a “manifold.” This word was brought into math by Bernhard Riemann, one of the most imaginative mathematicians that ever lived. In a sensational lecture in 1854 — it left Gauss himself stunned — Riemann laid out the key principles for investigating manifolds of any number of dimensions. He showed how to study a manifold not from the “outside,” as a shaped object imbedded in a space of higher dimension, but from within the space itself. Riemann’s brilliant insights were the foundation for all that followed.
Henri Poincaré was born just six weeks before that lecture, into a high-bourgeois French family. (Raymond Poincaré, president of France during World War I, was his first cousin.) As a mature mathematician working at the end of the 19th century, he built on Riemann’s foundations to create — “almost single-handedly,” says Mr. O’Shea — the mathematical topic called “algebraic topology,” which embraces both the local properties of a manifold (how flat or curved it is near any point) and the global ones (e.g. whether it is finite or infinite, and whether or not it has “doughnut holes”).
Poincaré’s work was presented in six papers, published between1895 and 1904. Near the end of the last paper occurs this remark:
There remains one question to handle: Is it possible that the fundamental group of a manifold could be the identity, but that the manifold might not be homeomorphic to the three-dimensional sphere? … But this question would carry us too far away.
That “one question” became famous as the Poincaré Conjecture. All through the 20th century, mathematicians struggled to resolve it. In 1961, Stephen Smale resolved the equivalent conjecture for five or more dimensions (it is not possible). In 1982, Michael Freedman did the same for four dimensions (same answer). The case of Poincaré’s three-dimensional manifolds, however — the kind of manifolds useful for modeling our three-dimensional space — remained open.
Then, in three papers published in 2002-3, the eccentric Russian mathematician Grigory Perelman claimed to have resolved Poincaré’s three-dimensional case. It took two years for the world’s mathematicians to satisfy themselves that Mr. Perelman’s reasoning was sound. They awarded him the coveted Fields Medal in 2006, but the reclusive Mr. Perelman declined it. Nor does he seem interested in collecting the Clay Institute’s \$1 million prize for resolving the conjecture. He lives quietly with his mother in her St. Petersburg flat, and may have abandoned mathematics altogether.
Donal O’Shea tells the whole story in this book, neatly interweaving his main theme with the history of ideas about our planet and universe. There is good coverage of all the main personalities involved, each one set in the social and academic context of his time. The author does his best to make the math accessible. The odder sorts of three-dimensional manifolds are by no means easy to visualize, even for professionals, but Mr. O’Shea offers some helpful analogies.
So what is the shape of our universe? The Poincaré Conjecture has little to say on this point. That is perfectly fair, since at present we know very little (though there are strong scientific reasons to be prejudiced against an infinite space). At this stage of our inquiries, the important thing is to fill the gaps in our understanding of three-dimensional manifolds in all their mathematical abstraction, so that we have a good range of well-understood mathematical models from which to draw. The resolution of the Poincaré Conjecture filled one of the most important gaps. Grigory Perelman deserves our gratitude, whether he wants it or not.