Geometry & Topology 13 (2009) 901–941

DOI: 10.2140/gt.2009.13.901

Abstract

We use the equivalence between embedded contact homology and Seiberg–Witten Floer homology to obtain the following improvements on the Weinstein conjecture. Let Y be a closed oriented connected 3–manifold with a stable Hamiltonian structure, and let R denote the associated Reeb vector field on Y . We prove that if Y is not a T²–bundle over S¹, then R has a closed orbit. Along the way we prove that if Y is a closed oriented connected 3–manifold with a contact form such that all Reeb orbits are nondegenerate and elliptic, then Y is a lens space. Related arguments show that if Y is a closed oriented 3–manifold with a contact form such that all Reeb orbits are nondegenerate, and if Y is not a lens space, then there exist at least three distinct embedded Reeb orbits.

Keywords

dynamical system, Seiberg–Witten, Floer homology

Mathematical Subject Classification

Primary: 53D40, 57R17, 57R57

Secondary: 57R58

References

Publication

Received: 21 September 2008
Revised: 8 December 2008
Accepted: 20 November 2008
Published: 8 January 2200
Proposed: Yasha Eliashberg
Seconded: Peter Ozsvath, Leonid Polterovich

Authors