Leafwise 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 any compact connected orientable manifold $M$ for which the foliation de Rham cohomology of the associated Reeb foliation has $H^0(F_{\lambda })\cong \mathbb{R}$. We also prove the same triviality for a generic choice of contact forms with fixed contact structure $\xi$. This vanishing result of $H^0(F_{\lambda })$ is also equivalent to the statement that 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 $\mathbb{R}$. On the other hand, we derive the rank of $H^1(F_{\lambda })$ is infinite whenever $\lambda$ admits a closed Reeb oribt, i.e., whenever Weinstein's conjecture holds. In particular we prove that $H^1(F_{\lambda })$ is infinite dimensional for all contact form $\lambda$ in dimension 3, thanks to Taubes' proof of 3-dimensional Weinstein's conjecture.

37 pages, comments welcome! v2) 55 pages, the case with fixed contact structure added, new perspective on contact dynamics added, exposition much improved; v3) 44 pages, the paper largely rewritten, exposition simplified, more references added, an error concerning the cokernel statement corrected and amplified