Stable Hamiltonian structures in dimension 3 are supported by open books (Q2921096)

A Hamiltonian structure (HS) on a closed oriented 3-manifold \(M\) is a closed nowhere zero 2-form \(\omega\) on \(M\). A stable Hamiltonian structure (SHS) on \(M\) is a pair \((\omega,\lambda)\) consisting of a closed 2-form \(\omega\) and a 1-form \(\lambda\) such that \(\lambda \wedge \omega >0\) and \(\mathrm{ker}(\omega) \subset\mathrm{ker}(d \lambda)\). The first part of the paper is devoted to the proof of two theorems which state that every open book supports a SHS realizing a given cohomology class and given signs at the binding components and that any two SHS supported by the same open book in the same cohomology class and with the same signs at the binding components are connected by a stable homotopy supported by the open book. The remainder of the paper is devoted to preliminary results for the proof of another theorem which states that every SHS on closed oriented 3-manifolds is stably homotopic to one which is supported by an open book. As a conclusion, various examples of SHS supported by open books on simple manifolds \(S^3\) and \(S^1 \times S^2\) are considered.