Spectral sequence of an isometric action

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 $G$.