Symplectic non-Hamiltonian circle actions on compact connected 4-manifolds with discrete period group are classified up to equivariant symplectomorphism by seven invariants.
The Kirby torus trick for surfaces
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abstract
This is an expository paper giving a proof of the existence and uniqueness of smooth structures (hence also PL structures) on topological surfaces. Most published proofs rely on the topological Schoenflies theorem, but here we use instead the Kirby torus trick. This has the advantage of reducing the point-set topology in the proof to practically nothing, replacing it by a few basic facts about smooth surfaces. Uniqueness of smooth structures is proved in the strong form that every homeomorphism between smooth surfaces is isotopic to a diffeomorphism.
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Classification of symplectic non-Hamiltonian circle actions on 4-manifolds
Symplectic non-Hamiltonian circle actions on compact connected 4-manifolds with discrete period group are classified up to equivariant symplectomorphism by seven invariants.