Round 5: Tossup 20

John Milnor constructed an “exotic” differentiable manifold that is (15[1])homeomorphic (15[1])but not diffeomorphic to these objects in dimension seven. Stereographic projection is used to map points on one of these objects (-5[1])to the complex (15[1])plane. The idea that (15[1])every simply connected (15[1])closed 3-manifold is homeomorphic to one of these surfaces (15[1])was the subject of the (*) Poincaré (“pwann-cah-RAY”) conjecture that (10[1])Grigory Perelman proved. A coordinate system named for these objects utilizes polar and azimuthal angles. For 10 points, name these objects, which contain all points equidistant from a center in three dimensions. ■END■ (10[1])

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