Giroux correspondence, confoliations, and symplectic structures on S^1 x M

TL;DR

This paper proves that if the product of a closed oriented 3-manifold with a circle admits a symplectic structure, then the manifold fibers over the circle, using Giroux's correspondence and properties of symplectic actions.

Contribution

It establishes that such 3-manifolds are fibered over S^1 by leveraging Giroux's correspondence and symplectic actions, connecting contact structures to fibered structures.

Findings

M is a fiber bundle over S^1 when S^1 x M admits a symplectic structure

The S^1-action on S^1 x M can be made symplectic

Uses Giroux's correspondence to relate contact structures and open book decompositions

Abstract

Let M be a closed oriented 3-manifold such that S^1 x M admits a symplectic structure w. The goal of this paper is to show that M is a fiber bundle over S^1. The basic idea is to use the obvious S^1-action on S^1 x M by rotating the first factor, and one of the key steps is to show that the S^1-action on S^1 x M is actually symplectic with respect to a symplectic form cohomologous to w. We achieve it by crucially using the recent result or its relative version of Giroux about one-to-one correspondence between open book decompositions of M up to positive stabilization and co-oriented contact structures on M up to contact isotopy.