The Conjecture was posed as a question in a paper that French mathematician Henri Poincaré wrote in 1904: “If a three-dimensional shape is simply connected, is it homeomorphic to the three-sphere?” The conjecture is fundamental to achieving an understanding of all three-dimensional shapes.
To understand the statement, consider the analogous two-dimensional situation. Think of a rubber band stretched around the spherical surface of an apple. It is easily seen that that it can be shrunk to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if the same rubber band were stretched around a doughnut, a torus, there would be no way of shrinking it to a point without bearing either the rubber band or the doughnut. In the language of Mathematics, the apple is “simply connected,” while the doughnut is not.
How do we characterise the spherical apple surface mathematically? Think of a disk lying in the two-dimensional plane. A sphere is nothing but the disk with its boundary lifted up and tied to a single point. Mathematically we say that all the boundary points are identified to a single point. Now one can do this only if the 2-dimensional disk is lying in a 3-dimensional space. So what we have is a “2-dimensional sphere in a 3-dimensional space.”
Stating that a 2-dimensional sphere is characterised by the property of simple connectivity, Poincaré's question corresponded to asking whether similar characterisation is valid for all closed 3-dimensional objects (embedded in a 4-dimensional space) that are sufficiently like a 3-dimensional sphere.
The question has turned out to be extraordinarily difficult and mathematicians have been struggling to prove this ever since. The analogous result has been known to be true in higher dimensions for some time now but the case of the three dimensional sphere has proved to be the hardest of all. It required a maverick Perelman to crack it.
In the box item that went with the main report, “It was a tough nut to crack”, a sentence in the second paragraph was “On the other hand, if the same rubber band were stretched around a doughnut, a torus, there would be no way of shrinking it to a point without bearing either the rubber band or the doughnut.” The word should have been “tearing”.