Round 5: Tossup 20

John Milnor constructed an “exotic” differentiable manifold that is homeomorphic but not diffeomorphic to these objects in dimension seven. Stereographic projection is used to map points on one of these objects to the complex plane. The idea that every simply connected closed 3-manifold is homeomorphic to one of these (15[1])surfaces was the subject of the (*) Poincaré (“pwann-cah-RAY”) conjecture (10[1])that Grigory Perelman proved. A coordinate system named for these objects utilizes polar and azimuthal angles. (10[1])For 10 points, name these objects, which contain all points equidistant from a center in three dimensions. ■END■

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