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.
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 Mbius 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.
The proof was easy to verify in one and two dimensions, where all possible manifolds were classified (Poincar contributed to the classification of 2-dimensional manifolds). Unfortunately, Poincar soon devised a second 3-dimensional manifold that had the same homology as the 3-sphere. His "proof" was false.
Undeterred, Poincar formulated a different measure, called homotopy. Homotopy works by imagining that you embed a closed loop in the manifold in question [see illustration]. The loop can be wound around the manifold in any possible fashion. We then ask, can the loop be shrunk down to a point, just by moving the loop around, without ever lifting a piece of it out of the manifold? On a shape like a doughnut the answer is no. If the loop runs around the circumference of the doughnut it cannot be shrunk to a point--it gets caught on the inner ring. Homotopy is a measure of all the different ways a loop can get caught.
On an n-sphere, no matter how convoluted a path the loop takes it can always be untangled and shrunk to a point. (The loop is allowed to pass through itself during these manipulations.) Poincar speculated that the only 3-manifold on which every possible loop can be shrunk to a point was the 3-sphere itself. This time Poincar knew he didn't have a proof, and he didn't venture any thoughts about dimensions higher than 3. In due course this proposal became known as the Poincar conjecture. Over the decades, many people have announced proofs of the conjecture, only to be proved wrong.
Copacabana and Higher Dimensions
A dramatic breakthrough in a thoroughly unexpected direction was achieved in 1960 by Steve Smale, who is now a professor emeritus at the University of California, Berkeley. Smale heard about the Poincar conjecture in 1955, while he was a graduate student at the University of Michigan in Ann Arbor. He soon found what he believed to be a proof, but it was yet another wrong proof.
In 1960, Smale spent 6 months as a post-doctoral fellow at the Institute of Pure and Applied Mathematics in Rio de Janeiro. He and his young family lived in an apartment near the beaches of Copacabana. In the mornings, Smale took pen and paper to the beach, where he swam, body surfed and did mathematics. In this idyllic setting, he came up with two major advances. ("My best known work," he later commented, "was done on the beaches of Rio de Janeiro.") The first advance related to the dynamics of systems that exhibit chaotic motion, a topic that Poincar had initiated. Although Poincar laid the foundations of chaos theory at the turn of the century, only in the 1970s did the topic acquire that name and take off as a (rediscovered) revolutionary concept. Smale's work with dynamical systems, which involve mappings from one manifold to another, suggested an angle for attacking the Poincar conjecture. The attack succeeded for manifolds of dimension 5 and higher, proving that the n-sphere is the unique, simplest manifold for five or more dimensions.
Not long after Smale's proof was announced, John R. Stallings (now also at Berkeley) used a completely different method to prove the conjecture for dimensions 7 and above. Christopher Zeeman (then at the University of Cambridge; now retired) extended this proof down to 5 dimensions.
In the four-dimensional case, the methods used by Smale, Stallings and Zeeman fail. But in 1982 Michael H. Freedman, now at Microsoft Research, succeeded in proving a 4-dimensional version of the Poincar conjecture (above three dimensions there are subtly different ways to formulate the conjecture). Many questions about the 4-dimensional case remain open today, including the truth of variants of the conjecture other than Freedman's.
4-manifolds have some astonishing complexities that are not present in either higher or lower dimensions. It is interesting to speculate that it is not a coincidence that our universe of space and time is four dimensional, the most mathematically complicated case. Perhaps only that case can encompass the complexity necessary for life. Or perhaps the physics of a universe is inherently driven toward the most complicated possibility.
The results of Smale and company represent information about what space is like, at its most fundamental level, for various dimensionalities. They tell us that for four dimensions or greater, the simplest space--the n-sphere--is unique. If you take any space of n dimensions, either you have the n-sphere or you have a space that has a structure around which a loop of thread can get "caught," like around the hole in a doughnut.
Although it may be tempting to think of higher dimensions as being just like lower dimensions with something extra, higher dimensions differ in many ways from their lower-order counterparts. For example, in three dimensions a loop of thread can form a knot that cannot be untied without cutting the thread. That is impossible in one or two dimensions--you can't even form a knot. In four or more dimensions, any "knot" of thread can be untied: you can use the extra dimension to move segments of the thread past one another. The existence of knots makes three-dimensional spaces vastly more complicated than two-dimensional ones, and quite different from higher-dimensional spaces, where every knot is equivalent to a simple closed loop. Yet the higher-dimensional Poincar conjectures tell us that with respect to the fundamental properties of interest to topologists, higher dimensions hold no surprises.
Graham P. Collins is a staff writer and editor.