# Poincaré conjecture

An assertion attributed to H. Poincaré and stating: Any closed simply-connected three-dimensional manifold is homeomorphic to the three-dimensional sphere. A natural generalization is the following assertion (the generalized Poincaré conjecture): Any closed $n$-dimensional manifold which is homotopy equivalent to the $n$-dimensional sphere $S^n$ is homeomorphic to it; at present (1991) it has been proved for all $n\geq5$ (and for smooth manifolds also when $n=4$).