For 100 years mathematicians have been trying to prove a conjecture that was first proposed by Henri Poincar¿ relating to an object known as the three-dimensional sphere, or 3-sphere. The conjecture singles out the 3-sphere as being unique among all three-dimensional objects, or manifolds. A proof of the Poincar¿ conjecture has finally come, with the work of a young Russian mathematician, Grigori Perelman. He could potentially win a $1-million prize from the Clay Mathematics Institute--the Poincar¿ conjecture is one of its seven "Millennium Problems." Perelman's analysis also completes a major research program that classifies all possible three-dimensional manifolds. Our universe might have the shape of a 3-sphere. The mathematics has other intriguing connections to particle physics and Einstein's theory of gravity.

All this is explored in greater detail in the July 2004 print edition of Scientific American. Here I focus on Poincar¿ himself and the early years of his conjecture, in particular the astonishing results that proved higher-dimensional versions of the conjecture in the latter half of the twentieth century.

Poincar¿
Henri Poincar¿ was one of the two foremost mathematicians who were active at the turn of the 20th century (the other being David Hilbert). He has been called the last universalist--one who was at ease in all branches of mathematics, both pure and applied. In addition to advancing numerous branches of mathematics Poincar¿ contributed to the theories of celestial mechanics and electromagnetism and to the philosophy of science (about which he wrote several widely read popular books; a youthful Albert Einstein and friends were greatly impressed by one of these). Along with these highly theoretical pursuits, Poincar¿ served as an engineer inspecting coal mines. He rose to be Inspector General of the Corps des Mines and the president of the French Bureau of Longitude, where he oversaw the precision mapping of the globe using the new techniques of time synchronization by undersea cables and telegraphy.

Poincar¿ independently discovered several key aspects of relativity before and concurrent with Einstein. In 1904, one year before Einstein published his landmark paper on relativity, Poincar¿ spoke prophetically at an international conference, "Perhaps we must construct a new mechanics, of which we can only catch a glimpse,... in which the velocity of light would become an impassable limit." The group of transformations in relativity theory that relate what two different observers see is now known as the Poincar¿ group.

Poincar¿ largely created the branch of mathematics called algebraic topology. Using techniques from that field, in 1900, Poincar¿ analyzed the properties of spheres in various dimensions. To a topologist, a circle (the rim of a disk, not the disk itself) is a "1-dimensional sphere," or a 1-sphere. The circle is 1-dimensional because it takes only one number to specify a location on the circle. A "2-sphere" is the shape of a spherical balloon. Two coordinates--latitude and longitude--are needed to specify a position on the balloon. The 3-sphere is the three-dimensional analogue of these and is described in detail in the print edition. Similarly, each dimensionality n has an n-sphere. Mathematicians call an object or a space of any dimensionality a manifold. The study of manifolds is called topology.

The 2-sphere is unique among all possible finite 2-dimensional manifolds: every other such manifold is more complicated and can be made from a 2-sphere by performing some combination of three operations: cutting out pieces, attaching "handles" (a shape just like the handle on a cup), or incorporating a strange twist. like the twist in a M¿bius band. Mathematicians were keenly interested to know if the n-sphere in dimensions 3 and up were similarly unique.

To tackle this question, Poincar¿ used a new measure of topological complexity called homology. Roughly speaking, homology detects how many cavities of different dimensions are enclosed by the manifold. But all that you need to know here is that the homology of a manifold specifies certain topological properties that it has. Poincar¿ proved that in each dimension n the only manifold having the homology of the n-sphere was the n-sphere itself.