Spherical space form conjecture

Heinz Hopf posed the question in 1926: every finite group action on the three-sphere should be conjugate to an isometric action. Perelman's verification of geometrization settled it, with the result also encompassing the elliptization and Poincaré conjectures.

Known results

• Smith's work (1944) handled actions with fixed points.
• Livesay and Myers treated cyclic groups of order a power of $2$.
• Rubinstein treated cyclic groups of order $3$.
• Maher and Rubinstein proved that every free action of $\mathbb{Z}_3$ on $S^3$ is standard.

Perelman’s geometrization proof, 2005–2006

Perelman's work, building on Hamilton's Ricci flow, established geometrization and thereby showed that closed $3$-manifolds with finite fundamental group are spherical space forms. Thus the stated $3$-sphere action conjecture is a theorem; no credible recent counterclaim or unresolved objection was found.

Current status (as of August 2026): The conjecture is resolved as a consequence of Perelman’s geometrization theorem; no case of the stated three-dimensional assertion remains open.