Generalized Poincaré conjecture

Remarks

For $n \le 2$ the statement follows from the classification of curves and surfaces. For $n \ge 5$ it was proved by Stephen Smale for smooth manifolds, through handle decompositions and the h-cobordism theorem. Max Newman extended it to topological manifolds, and John Stallings and Christopher Zeeman gave other proofs. For $n = 4$ it was proved by Michael Freedman in 1982, and for $n = 3$ it is the Poincaré conjecture, proved by Grigori Perelman with Ricci flow. The analogous statement with "diffeomorphic" in place of "homeomorphic" is false: John Milnor found smooth manifolds homeomorphic but not diffeomorphic to $S^7$, the exotic spheres. The smooth version for $n = 4$ is an open problem. For $n \ge 2$, a closed $n$-manifold is homotopy equivalent to $S^n$ as soon as it is simply connected and has the integer homology of $S^n$, by the Hurewicz and Whitehead theorems.

Going deeper

For a student

The theorem is the starting point of the classification of manifolds up to homeomorphism. In dimensions at least five, it comes out of the h-cobordism theorem, which turns homotopy information into homeomorphisms using handle moves. In the smooth category the question becomes the study of exotic spheres, which Kervaire and Milnor organized into groups of homotopy spheres under connected sum.