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Spherical space form conjecture

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In mathematics, the spherical space form conjecture in geometric topology states that a finite group acting on the 3-sphere is conjugate to a group of isometries of the 3-sphere acting by left translation.

Currently the conjecture is known for groups whose actions have fixed points -- this special case is known as the Smith conjecture. It is also known for various groups acting without fixed points, such as cyclic groups whose orders are a power of two (Livesay, Myers) and cyclic groups of order 3 (Rubinstein). Thurston's Elliptization conjecture implies the spherical space form conjecture in all cases.

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