On the equivariant cohomology of homogeneous spaces.
Abstract: 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... read moreoriginal 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
Thesis (Ph.D.)--Tufts University, 2015.
Submitted to the Dept. of Mathematics.
Advisor: Loring Tu.
Committee: Julianna Tymoczko, George McNinch, and Fulton Gonzalez.
Keyword: Mathematics.read less