Foliation de Rham cohomology of generic Reeb foliations

Abstract

In this paper, we prove that there exists a residual subset of contact forms $\lambda$ (if any) on a compact connected orientable manifold $M$ for which the foliation de Rham cohomology of the associated Reeb foliation $F_\lambda$ is trivial in that both $H^0(F_\lambda,{\mathbb R})$ and $H^1(F_\lambda,{\mathbb R})$ are isomorphic to $\mathbb R$. We also prove the same triviality for a generic choice of contact forms with fixed contact structure $\xi$. For any choice of $\lambda$ from the aforementioned residual subset, this cohomological result can be restated as any of the following two equivalent statements: (1) The functional equation $R_{\lambda}[f] = u$ is uniquely solvable (modulo the addition by constant) for any $u$ satisfying $\int_M u\, d\mu_\lambda =0$, or (2) The Lie algebra of the group of strict contactomorphisms is isomorphic to the span of Reeb vector fields, and so isomorphic to the 1 dimensional abelian Lie algebra $\mathbb R$. This result is also a key ingredient for the proof of the generic scarcity result of strict contactomorphisms by Savelyev and the author.