# Cyclic actions on rational ruled symplectic four-manifolds

## TL;DR

This paper proves that on rational ruled symplectic four-manifolds, any homologically trivial cyclic symplectic action can be extended to a Hamiltonian circle action, revealing a strong link between cyclic and circle symmetries.

## Contribution

It establishes that all homologically trivial cyclic symplectic actions on rational ruled four-manifolds are restrictions of Hamiltonian circle actions, a novel classification result.

## Key findings

- Homologically trivial cyclic actions are restrictions of Hamiltonian circle actions.
- The result applies specifically to rational ruled symplectic four-manifolds.
- Provides a classification of symplectic actions in this setting.

## Abstract

Let $(M,\omega)$ be a ruled symplectic four-manifold. If $(M, \omega)$ is rational, then every homologically trivial symplectic cyclic action on $(M,\omega)$ is the restriction of a Hamiltonian circle action.