The topology of symplectic circle bundles

TL;DR

This paper characterizes when circle bundles over three-manifolds with symplectic total spaces admit invariant symplectic forms, revealing topological constraints on the base and total space.

Contribution

It establishes conditions under which symplectic circle bundles over three-manifolds admit invariant symplectic forms, linking topology and symplectic geometry.

Findings

Base must be irreducible or S^2 x S^1

Invariant symplectic form exists under specific conditions

Links between Seifert fibered bases, Thurston norm, and Lefschetz fibrations

Abstract

We consider circle bundles over compact three-manifolds with symplectic total spaces. We show that the base of such a space must be irreducible or the product of the two-sphere with the circle. We then deduce that such a bundle admits a symplectic form if and only if it admits one that is invariant under the circle action in three special cases: namely if the base is Seifert fibered, has vanishing Thurston norm, or if the total space admits a Lefschetz fibration.