Round 5: Tossup 20

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

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