Spherical space form conjecture

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

The conjecture is implied by Thurston's geometrization conjecture, which was proven by Grigori Perelman in 2003. The conjecture was independently proven for groups whose actions have fixed points—this special case is known as the Smith conjecture. It is also proven 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).

References

• Hass, Joel (2005), "Minimal surfaces and the topology of three-manifolds", Global theory of minimal surfaces, Clay Math. Proc. 2, Providence, R.I.: Amer. Math. Soc., pp. 705–724, MR 2167285