Poincaré conjecture

Remarks

Poincaré asked this question in 1904. Perelman proved it in 2002 and 2003, following Hamilton's program of Ricci flow. One runs the Ricci flow from any Riemannian metric on $M$ and cuts out regions where the curvature blows up (surgery). For simply connected $M$ the flow with surgery becomes extinct in finite time, and this shows that $M$ is a sphere. It is also a special case of the geometrization conjecture. Both hypotheses matter: $\R^3$ is simply connected but not closed, and the Poincaré homology sphere is closed with the homology of $S^3$ but has nontrivial fundamental group. The analogous statement in dimensions $n \ge 5$ is due to Smale and in dimension $4$ to Freedman, in their topological versions.

Going deeper

For a student

The statement is a recognition theorem: a single algebraic condition identifies $S^3$ among all closed $3$-manifolds. In practice it is now used as part of geometrization, which gives every closed $3$-manifold with finite fundamental group the form $S^3 / \Gamma$ for a finite group $\Gamma$ of isometries. In the smooth category, it also follows that every smooth homotopy $3$-sphere is diffeomorphic to $S^3$, because topological $3$-manifolds have unique smooth structures.