Round 5: Tossup 20

John Milnor constructed an “exotic” differentiable manifold that is homeomorphic but not diffeomorphic to these objects in (-5[1])dimension seven. Stereographic projection (15[1])is used to map points on one of these objects (15[1])to the complex plane. The idea that every simply connected closed 3-manifold is homeomorphic to one (15[1]-5[1])of these surfaces was the subject (15[1])of the (*) Poincaré (“pwann-cah-RAY”) conjecture that Grigory Perelman proved. A coordinate system (10[1])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■ (0[1])

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