Smith conjecture

Remarks

P. A. Smith showed that the fixed point set of such a map of prime order is empty or a circle, and asked whether the circle can be knotted. The proof combined work of several authors, including Thurston on hyperbolic structures, Meeks and Yau on minimal surfaces, Bass on group theory, and Gordon and Litherland. One case uses incompressible surfaces in the knot complement; the other uses a hyperbolic structure given by Thurston's work. The orientation hypothesis matters: a reflection of $S^3$ has order $2$ and fixes a $2$-sphere. The analogue in higher dimensions fails: in $S^4$, some knotted $2$-spheres are fixed sets of periodic diffeomorphisms. Example: the rotation by angle $2\pi/n$ about the unknotted great circle $\set{(z, 0)} \subset S^3 \subset \C^2$, given by $(z, w) \mapsto (z, e^{2\pi i/n} w)$, fixes exactly that circle.

Going deeper

For a student

Equivalently, a nontrivial knot $K$ is never the branch set of a cyclic branched cover $S^3 \to S^3$. The theorem was a test case for Thurston's geometrization program: the decisive input came from hyperbolic geometry and minimal surfaces. It is a precursor of the orbifold theorem, which handles more general finite group actions on $3$-manifolds.