Leafwise de Rham cohomology of generic Reeb foliations

2025/04/23 by Yong-Geun Oh, Yong‐Geun Oh, Oh, Yong-Geun · 1 voice
Mathematics · #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #math.SG

paper · pdf · doi:10.48550/arxiv.2504.16453

openalex publication_date 2025/04/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we prove that there exists a residual subset of contact forms λ (if any) on any compact connected orientable manifold M for which the foliation de Rham cohomology of the associated Reeb foliation has H⁰(F_λ) ≅ \mathbb R. We also prove the same triviality for a generic choice of contact forms with fixed contact structure ξ. This vanishing result of H⁰(F_λ) 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 H¹( F_λ) is infinite whenever λ admits a closed Reeb oribt, i.e., whenever Weinstein's conjecture holds. In particular we prove that H¹(F_λ) is infinite dimensional for all contact form λ in dimension 3, thanks to Taubes' proof of 3-dimensional Weinstein's conjecture.

Discussions

• Yong-Geun Oh: Foliation de Rham cohomology of generic Reeb foliations https://arxiv.org/abs/2504.16453 https://arxiv.org/pdf/2504.16453 https://arxiv.org/html/2504.16453 [bsky, 0 points, 1 comments]

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