Foliation de Rham cohomology of generic Reeb foliations
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916731324203008 |
|---|---|
| author | Oh, Yong-Geun |
| author_facet | Oh, Yong-Geun |
| contents | In this paper, we prove that there exists a residual subset of contact forms $λ$ (if any) on a compact connected orientable manifold $M$ for which the foliation de Rham cohomology of the associated Reeb foliation $F_λ$ is trivial in that both $H^0(F_λ,{\mathbb R})$ and $H^1(F_λ,{\mathbb R})$ are isomorphic to $\mathbb R$. We also prove the same triviality for a generic choice of contact forms with fixed contact structure $ξ$. For any choice of $λ$ from the aforementioned residual subset, this cohomological result can be restated as any of the following two equivalent statements: (1) The functional equation $R_λ[f] = u$ is uniquely solvable (modulo the addition by constant) for any $u$ satisfying $\int_M u\, dμ_λ=0$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_16453 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Foliation de Rham cohomology of generic Reeb foliations Oh, Yong-Geun Symplectic Geometry 37C86, 53D10, 53D35, 37C05 In this paper, we prove that there exists a residual subset of contact forms $λ$ (if any) on a compact connected orientable manifold $M$ for which the foliation de Rham cohomology of the associated Reeb foliation $F_λ$ is trivial in that both $H^0(F_λ,{\mathbb R})$ and $H^1(F_λ,{\mathbb R})$ are isomorphic to $\mathbb R$. We also prove the same triviality for a generic choice of contact forms with fixed contact structure $ξ$. For any choice of $λ$ from the aforementioned residual subset, this cohomological result can be restated as any of the following two equivalent statements: (1) The functional equation $R_λ[f] = u$ is uniquely solvable (modulo the addition by constant) for any $u$ satisfying $\int_M u\, dμ_λ=0$, 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. |
| title | Foliation de Rham cohomology of generic Reeb foliations |
| topic | Symplectic Geometry 37C86, 53D10, 53D35, 37C05 |
| url | https://arxiv.org/abs/2504.16453 |