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

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

2008-11-05

arxiv.org/abs/0811.0641v3

Mathematics

Geometric Topology

17 pages; Sec.3 rewritten for more clarity

Scientific paper

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.

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