Henri Poincaré (French mathematician, 1854 – 1912) posed the question in 1904 whether any three dimensional surface with a few specific properties was in some mathematical sense equivalent to the three dimensional sphere. Poincaré’s famous conjecture is an excellent example of a general methodology in mathematics: start with a set of objects; put some type of structure on that set; define what it means for two different sets to be equivalent with that structure; then start identifying equivalent sets. Sometimes there are surprisingly few canonical examples that represent all possible sets with these structures, i.e. all possible sets with a given structure are actually equivalent to one of only a few examples.
In Poincaré’s conjecture, the set is a three dimensional surface of points living in four or more dimensions (or at least the mathematical generalization of such a surface – called a “manifold”). Two such surfaces are said to be equivalent if there exists an invertible continuous function from one surface onto the other. The intuition behind this equivalence is that there is a one-to-one identification between points on the former surface with points on the latter, and that this identification preserves distances in the sense that points that are close together on one surface are identified with points that are close together on the other. Trying to pin down this notion of “close” and what is means to preserve distance is what makes the mathematics start to get quite complicated. For the Poincaré’ Conjecture, the notion of equivalence that seems to matter is “genus”, or “number of holes”.
As an example, consider a beach ball – a two dimensional sphere. If you were to trace a path on the sphere without picking up your pencil, without crossing over the path anywhere, and ending where you started, you’d have something called a “closed, simple loop”. Now notice that on the sphere, you can always shrink these loops down to a point without leaving the sphere. For example, any line of latitude (one example of a closed, simple loop) can be “pushed” to a pole, where it shrinks to that pole without leaving the sphere. In contrast, consider a doughnut – a torus. If you were to draw a path on the torus going around the hole in the center, you can convince yourself that it’s impossible to shrink that path down to a point while staying on the torus. That path is stuck being a loop: you can’t shrink it out of exist while staying on the torus. The sphere has genus 0, and the torus has genus 1. They are fundamentally different this way.
Since we now understand some of the mathematical concepts involved, we’re in a position to state Poincaré’s Conjecture more specifically: every closed three dimensional manifold of genus 0 is equivalent to the 3 dimensional sphere. (Here, “closed” means it doesn’t go on forever, and it has no edges that are left out. Both of the above examples, the sphere and the torus, are closed manifolds. A plane, by contrast, goes on forever – it’s not closed.) This is known to be the case for two dimensional surfaces (genus is enough to determine equivalence); and it’s known not to be case for four or more dimensions (genus alone does not determine equivalence – there’s just too much room for other things to happen), but it remained unknown for three dimensional surfaces until 2003, when it was finally solved by the Russian mathematician Grigori Perelman. He showed that it’s true: any such manifold is equivalent to the 3 dimensional sphere.
Perelman refused the many honors and prizes offered for settling the Poincaré Conjecture. He says his contribution was no greater than that of other mathematicians who had done much of the preliminary work.
