The Orbit Structure of a G-Space X and the Ideal Theoretical Invariants of H ^* _G (X)

Abstract

In this chapter, we shall proceed to investigate the relationship between the geometric structures of a given G-space X and the algebraic structures of its equivariant cohomology H ^* _G (X). From the viewpoint of transformation groups, those structures which are usually summarized as the orbit structure are certainly the most important geometric structures of a given G-space. Hence, it is almost imperative to investigate how much of the orbit structure of a given G-space X can actually be determined from the algebraic structure of its equivariant cohomology H ^* _G (X). To be more precise, let us formulate a few more specific problems as examples:

• Problem 1. How much of the cohomology structure of the fixed point set, H*(F), is determined by the equivariant cohomology H ^* _G (X)?

• Problem 2. Is it possible to give a criterion for the existence of fixed points purely in terms of the equivariant cohomology H ^* _G (X)?

• Problem 3. Suppose F(H, X) = Ø. How to determine the set of maximal isotropy subgroups, {H_i ⊂ G; maximal among those H with F(H, X) # Ø} from the algebraic structure of H ^* _G (X)?