Transformation Groups on Compact Homogeneous Spaces

Abstract

The existence of abundant linear actions and the simplicity of topological structure form exactly the ideal combination that makes euclidean spaces, disks, spheres and projective spaces the best testing spaces for the study of transformation groups. So far, most of the deep results in topological transformation groups are still largely concentrating in the study of the above testing spaces. Generally speaking, the ideal combination of topological simplicity and abundant linear actions of the testing spaces are certainly very helpful in obtaining some basic understandings to begin with. For example, it is exactly the classical linear representation theory and those specific results of Chapters V and VI concerning such testing spaces that lead us to the basic understanding of the central importance of elementary abelian groups in the whole theory of topological transformation groups as well as to the formulation of those fundamental splitting theorems of Chapter IV (for actions of elementary abelian groups) in the setting of equivariant cohomology theory. However, the perspective will be undesirably limited if one becomes self-indulgent in insisting on the cohomological simplicity of the testing spaces. In this chapter, we begin to broaden the domain of testing spaces by considering transformation groups on compact homogeneous spaces. Since compact homogeneous spaces cover a wide range of topological types but still accommodate a rich variety of natural actions, the are particularly suitable for the study of transformation group.