Construction of symplectic structures on 4-manifolds with a free circle action

TL;DR

This paper develops a method to construct symplectic structures on certain 4-manifolds with free circle actions, extending previous work and fully characterizing the symplectic cone for product manifolds like S^1×N.

Contribution

It generalizes earlier constructions of symplectic forms on 4-manifolds with circle actions and completely determines the symplectic cone for product manifolds S^1×N.

Findings

Constructs symplectic forms on 4-manifolds with circle actions under fibering conditions.

Extends Thurston, Bouyakoub, and Fernandez-Gray-Morgan's work on symplectic structures.

Fully determines the symplectic cone for product 4-manifolds S^1×N.

Abstract

Let $M$ be a closed 4-manifold with a free circle action. If the orbit manifold $N^3$ satisfies an appropriate fibering condition, then we show how to represent a cone in $H^2(M;\R)$ by symplectic forms. This generalizes earlier constructions by Thurston, Bouyakoub and Fern\'andez-Gray-Morgan. In the case that $M$ is the product 4-manifold $S^1\times N$ our construction complements the results of \cite{FV08} (arXiv:0805:1234 [math.GT]) and allows us to completely determine the symplectic cone of such 4-manifolds. The content of this paper partly overlaps with the content of the unpublished preprint "Symplectic 4-manifolds with a free circle action" (arXiv:0801.1313 [math.GT]).