A Spectral Sequence for Equidimensional Actions of Compact Lie Groups

Abstract

In this article we provide a version of the Leray-Serre spectral sequence for equidimensional (i.e. smooth with all orbits of the same dimension) actions of compact connected Lie groups on compact manifolds. The main part of this article consists of the proof of the description of the second page of said spectral sequence. This description provides a link between the cohomology of the orbit space (basic cohomology of the foliation by orbits) the Lie algebra cohomology of the appropriate pair $(\mathfrak{g},\mathfrak{h})$ representing the cohomology of a generic orbit and the de Rham cohomology of the manifold. Due to the somewhat technical nature of the general description we have provided in the penultimate section a thorough study of special cases in which the sequence can be greatly simplified. In particular, vast simplifications can be obtained if the manifold $M$ on which the group acts is assumed to be simply connected or if the acting Lie group has some nice properties. In the final section, we show how to use a blow up process to use our sequence when the action is not equidimensional. We apply this method to give a topological obstruction to the existence Lie group actions on certain manifolds.