Classification of symplectic non-Hamiltonian circle actions on 4-manifolds

2024/11/15 by Rei Henigman, Henigman, Rei
Computer Science · Mathematics · #Geometric and Algebraic Topology #Hamiltonian (control theory) #Mathematical Dynamics and Fractals #Mathematical optimization #Mathematics #Pure mathematics #Symplectic geometry #Symplectic manifold #Symplectomorphism #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2411.10157

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2024/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We classify symplectic non-Hamiltonian circle actions on compact connected symplectic 4-manifolds, up to equivariant symplectomorphisms. Namely, we define a set of invariants, show that the set is complete, and determine which values are attainable by constructing a space for each valid choice. We work under the assumption that the group of periods of the one-form ι_X ω is discrete, which allows us to define a circle-valued Hamiltonian for the action, and apply tools from Karshon-Tolman's work on the classification of complexity one spaces. This assumption is always satisfied if the symplectic form is rational, or if the quotient space has first Betti number one.

Related

• Circle actions on symplectic four-manifolds
• A compact symmetric symplectic non-Kaehler manifold
• Symplectic 4-manifolds with fixed point free circle actions
• Non-symplectic smooth circle actions on symplectic manifolds
• Periodic symplectic and Hamiltonian diffeomorphisms on irrational ruled surfaces