Stable Hamiltonian structure and basic cohomology

Abstract

Let $(\omega , \lambda )$ be a stable Hamiltonian structure on a closed oriented manifold M of dimension $2n-1$, ${\mathcal {F}}$ the stable Hamiltonian foliation, generated by the Reeb vector field R of $(\omega , \lambda )$, and $H_{B}^{k}(M, {\mathcal {F}})$, the kth basic cohomology group of $(M, {\mathcal {F}})$; see Section 1 for definitions. In this paper, we give some topological properties of $(\omega , \lambda )$. In particular, we prove the following results:

• For all $k\in \{1, \ldots , n-1\}$,

  $$\begin{aligned} 0\ne [\omega ^{k}]\in H_{B}^{2k}(M, {\mathcal {F}}), \end{aligned}$$

  which allows us to give an example of a manifold with a Hamiltonian structure, which is not stable.

• If dim$H^{2}_{B}(M, {\mathcal {F}}) = 1$, then M is a co-symplectic manifold (symplectic mapping torus), or contact manifold.