Periodic symplectic and Hamiltonian diffeomorphisms on irrational ruled surfaces

TL;DR

This paper investigates the extension properties of symplectic and Hamiltonian cyclic actions on irrational ruled 4-manifolds, revealing cases where extensions are impossible and classifying certain group actions.

Contribution

It constructs examples of symplectic involutions that cannot extend to circle actions and classifies finite symplectic groups acting trivially on homology.

Findings

Constructed symplectic involutions not extendable to circle actions.

Proved that certain cyclic actions always extend to circle actions.

Classified finite groups acting trivially on first homology.

Abstract

We study the extension of homologically trivial symplectic or Hamiltonian cyclic actions to Hamiltonian circle actions on irrational ruled symplectic $4$-manifolds. On one hand, we construct symplectic involutions on minimal irrational ruled $4$-manifolds that cannot extend to a symplectic circle action even with a possibly different symplectic form. Higher dimensional examples are also constructed. On the other hand, for homologically trivial symplectic cyclic actions of any other order, we show that such an extension always exists. We also classify finite groups of symplecticmorphisms that acts trivially on the first homology group, and prove the non-extendability of the Klein $4$-group action to the three dimensional rotation group action motivated by the classification of finite groups of symplectomorphisms.