The Poincaré Conjecture

What just happened?

On a sphere, every loop drawn on it (an elastic "rubber band" lying on the surface) can be continuously contracted to a point, without leaving the surface and without tearing it. On a torus — if the loop is wrapped around the "hole" — this is impossible: the loop can only shrink to the edge of the hole, where it gets stuck, taut, and cannot be reduced any further.

The invariant — the number of "holes" (genus):

For a closed, smooth 2D surface, the classical formula for the Euler characteristic holds:

where $g$ — the genus (number of "holes"/handles). A sphere has $g=0$ (every loop contracts to a point), a torus has $g=1$ (not every loop contracts any more). It is precisely this number that distinguishes the two surfaces in the scene.

The true Poincaré Conjecture (1904, the 3D case):

Henri Poincaré asked: must every closed, "simply connected" (without holes, in the topological sense) three-dimensional manifold be topologically a sphere $S^3$? Formally:

— if the fundamental group of a closed three-dimensional manifold $M$ is trivial (that is, every loop contracts to a point), then $M$ is topologically "the same" as the three-dimensional sphere. In 2D this is a classical theorem — in 3D it turned out to be one of the hardest problems in mathematics.

How did Grigori Perelman prove it?

The problem stayed open for almost 100 years and was placed on the list of seven "Millennium Prize Problems" of the Clay Mathematics Institute — a one-million-dollar prize was set for solving each. In 2002–2003 the Russian mathematician Grigori Perelman published a series of papers in which he developed the technique of "Ricci flow" by Richard Hamilton — a tool that gradually "smooths" the geometry of a manifold, revealing its true topological nature. After years of detailed verification by the mathematical community, the proof was accepted as correct.

Declining the prizes

In 2006 Perelman was awarded the Fields Medal for this achievement ("for his contributions to geometry and his revolutionary insights into the analytic and geometric structure of the Ricci flow") — Perelman declined to accept it. In 2010 he also declined the one-million-dollar Clay Institute prize, after which he withdrew from mathematical and public life. The Poincaré Conjecture is so far the only solved one of the seven Millennium Prize Problems.

Why you need this for the exam:

The Poincaré Conjecture itself goes far beyond the school curriculum, but classifying surfaces by genus and the Euler characteristic $\chi = 2-2g$ is a good example of topology — the branch of mathematics in which what matters is not shape or dimensions, but whether something can be continuously deformed into something else without tearing.

Remember: the contractibility of every loop to a point is a property exclusive to surfaces without "holes" (genus 0, topologically spheres). Poincaré posed this question for the harder three-dimensional case in 1904 — Grigori Perelman solved it in 2002–2003, after which he declined both the Fields Medal and the one-million-dollar prize.