A closed simply connected three-manifold is homeomorphic to the three-sphere. Now a theorem, this identifies every homotopy three-sphere with the standard sphere and removes homotopy-sphere ambiguities in three-dimensional connected-sum classification. Smooth structures are also unique in dimension three.
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The Poincaré conjecture is a fundamental question in the field of topology, particularly in the study of three-dimensional spaces. Formulated by the French mathematician Henri Poincaré in 1904, the conjecture states that: **Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.
