Quantum invariants and finite group actions on three-manifolds

A 3-manifold M is said to be p-periodic (p⩾2 an integer) if and only if the finite cyclic group of order p acts on M with a circle as the set of fixed points. This paper provides a criterion for periodicity of rational homology three-spheres. Namely, we give a necessary condition for a rational homology three-sphere to be periodic with a prime period. This condition is given in terms of the quantum SU(3) invariant. We also discuss similar results for the Murakami–Ohtsuki–Okada invariant.