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

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.

17 pages; Sec.3 rewritten for more clarity

References in corpus (4)

• Géométrie de contact: de la dimension trois vers les dimensions supérieures
• Lectures on open book decompositions and contact structures
• Addendum to: "Knots, sutures and excision"
• Seiberg-Witten Floer homology and symplectic forms on S^1 X M^3