Spectral sequence of an isometric action

TL;DR

This paper provides a simplified proof of the spectral sequence associated with an isometric action of a compact Lie group on a manifold, extending the result to locally free actions with non-compact groups under certain conditions.

Contribution

It offers a straightforward proof of the second page of the spectral sequence for isometric group actions, extending to locally free actions with non-compact groups via an extension to a compact group.

Findings

The second page of the spectral sequence is given by the tensor product of the basic cohomology of the orbit space and the Lie algebra cohomology.

The proof avoids complex methods like Mayer-Vietoris or harmonic operators.

The result applies to locally free actions extended to a compact group.

Abstract

We consider a free smooth action $\Phi \colon G \times M \to M$ 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 $\Phi$ 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 $G$.