| The first part of this dissertation develops foundational material on the rational cohomology of Lie groups, their classifying spaces, and homogeneous spaces. In parallel, it develops the basics of Borel equivariant cohomology, with an aim to understanding equivariant cohomology of isotropy actions of K on compact homogeneous spaces G/K.;In the last few chapters, we establish several original results on such actions. Briefly, this work essentially reduces the question of when such an action is equivariantly formal to the case the isotropy subgroup K is a torus and the transitively acting group G is simply-connected, then completely classifies the possibilities in the event K further is a circle.;The appendices include an exposition of Borel's original proof of a theorem of Chevalley providing a framework for computing the cohomology of principal bundles, a computer program providing verification of a computationally intensive claim in the last chapter, and some applications, in fact the original motivation for this work, of the Berline--Vergne/Atiyah--Bott localization theorem to classical (pre-1941) results in topology.;A more detailed account of the content, including a delineation of what is original to this work and what is expository, can be found in the introduction. |
