Spectral sequence of an isometric action
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2026
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866918316443959296 |
|---|---|
| author | Prieto, J. I. Royo Saralegi-Aranguren, M. |
| author_facet | Prieto, J. I. Royo Saralegi-Aranguren, M. |
| contents | We consider a free smooth action $Φ\colon G \times M \to M$ of a connected compact Lie group $G$ on a manifold $M$. We examine the Cartan filtration of the complex of differential forms of $M$. The associated spectral sequence ${E}^{p,q}_{_{r}}$ converges to the cohomology of $M$. It is well known that the second page ${E}^{p,q}_{_{2}}$ of this spectral sequence is given by $H^{^p} (M/G) \otimes H^{^q} (\mathfrak g)$, where $\mathfrak g$ denotes the Lie algebra of $G$.
In this note, we provide a straightforward proof of this fact without using Mayer-Vietoris, harmonic operators, or other such methods found in existing proofs. In fact, we extend this result to the case where the action is locally free and $G$ is not compact, under the hypothesis that $Φ$ extends to a smooth action of a compact Lie group $K$. The compactness of $K$ is a crucial aspect of our proof.
When $G$ is not compact, the cohomology $H^{^p} (M/G) $ is not the cohomology of the orbit space $M/G$, which may be a topologically wild space, but rather the basic cohomology of the foliation determined by the action of $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_00271 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Spectral sequence of an isometric action Prieto, J. I. Royo Saralegi-Aranguren, M. Algebraic Topology Primary 57S15, Secondary 55N33 We consider a free smooth action $Φ\colon G \times M \to M$ of a connected compact Lie group $G$ on a manifold $M$. We examine the Cartan filtration of the complex of differential forms of $M$. The associated spectral sequence ${E}^{p,q}_{_{r}}$ converges to the cohomology of $M$. It is well known that the second page ${E}^{p,q}_{_{2}}$ of this spectral sequence is given by $H^{^p} (M/G) \otimes H^{^q} (\mathfrak g)$, where $\mathfrak g$ denotes the Lie algebra of $G$. In this note, we provide a straightforward proof of this fact without using Mayer-Vietoris, harmonic operators, or other such methods found in existing proofs. In fact, we extend this result to the case where the action is locally free and $G$ is not compact, under the hypothesis that $Φ$ extends to a smooth action of a compact Lie group $K$. The compactness of $K$ is a crucial aspect of our proof. When $G$ is not compact, the cohomology $H^{^p} (M/G) $ is not the cohomology of the orbit space $M/G$, which may be a topologically wild space, but rather the basic cohomology of the foliation determined by the action of $G$. |
| title | Spectral sequence of an isometric action |
| topic | Algebraic Topology Primary 57S15, Secondary 55N33 |
| url | https://arxiv.org/abs/2602.00271 |