The topology of symplectic circle bundles
Jonathan Bowden

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.
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Taxonomy
TopicsGeometric and Algebraic Topology · Homotopy and Cohomology in Algebraic Topology · Geometry and complex manifolds
