Spectral sequence of an isometric action

TL;DR

The paper proves that for a locally free smooth action of a connected Lie group $G$ on a manifold $M$, extended to a compact group $K$, the second page of the Cartan filtration LSdR spectral sequence satisfies $E^{p,q}_{2} \cong H^{p}(M/G) \otimes H^{q}(\mathfrak{g})$, where $H^{p}(M/G)$ is the basic cohomology of the orbit foliation. The authors extend this result to locally free, noncompact $G$ under the assumption that the action extends smoothly to a compact $K$, with the compactness of $K$ playing a critical role. Central to the approach is restricting to invariant forms via a good metric, establishing an invariant Cartan filtration, and using a projection operator $\rho$ to relate the full LSdR complex to the invariant one. The second-page computation is achieved by identifying $H^{*}(\mathfrak{g})$ with $(\wedge^{*} \mathfrak{g}^*)^{G}$ and constructing an isomorphism $F([\alpha] \otimes \beta) = \mathrm{class}(\alpha \wedge \beta)$, yielding the canonical tensor product structure on $E^{p,q}_{2}$ and linking basic foliated cohomology with Lie algebra cohomology.

Abstract

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 $G$.

Theorems & Definitions (29)

• Proposition 2.1
• proof
• Definition 3.1
• Definition 3.2
• Proposition 3.3
• proof
• Proposition 3.4
• proof
• Proposition 3.5
• proof