The Splitting Theorems and the Geometric Weight System of Topological Transformation Groups on Cohomology Projective Spaces

Abstract

In this chapter, we take those manifolds of the cohomology type of projective spaces as the testing spaces for the study of topological transformation groups. From the cohomological point of view, the projective spaces certainly have the simplest, and yet non-trivial, cohomology algebras, namely, truncate polynomial rings. Geometrically, the so-called projective transformation groups which are induced by the linear transformation groups still provide abundant interesting examples that we shall again call them “linear models”. In other words, projective spaces, endowed with a simple cohomology structure and an abundance of transformation groups, provide the ideal setting for the study of the cohomology theory of transformation groups.