A Spectral Sequence for locally free isometric Lie group actions

TL;DR

This paper introduces a spectral sequence connecting de Rham, Lie algebra, and basic cohomology for manifolds with free isometric Lie group actions, offering new computational tools and generalizations.

Contribution

It develops a novel spectral sequence for locally free isometric Lie group actions and introduces a new approach to de Rham cohomology in this context.

Findings

Provides a spectral sequence relating different cohomologies

Introduces a new description of de Rham cohomology for these actions

Generalizes the Wang long exact sequence to low codimension Lie algebra actions

Abstract

We present a spectral sequence for free isometric Lie algebra actions (and consequently locally free isometric Lie group actions) which relates the de Rham cohomology of the manifold with the Lie algebra cohomology and basic cohomology (intuitively the cohomology of the orbit space). In the process of developing this sequence we introduce a new description of the de Rham cohomology of manifolds with such actions which appears to be well suited to this and similar problems. Finally, we provide some simple applications generalizing the Wang long exact sequence to Lie algebra actions of low codimension.