Equivariant Cohomology of S2 Under Rotation

Abstract

This chapter shows how to use the spectral sequence of a fiber bundle to compute equivariant cohomology. As an example, it computes the equivariant cohomology of S2 under the action of S1 by rotation. The method of the chapter only gives the module structure of equivariant cohomology. Suppose a topological group G acts on the left on a topological space M. Let EG → BG be a universal G-bundle. The homotopy quotient MG fits into Cartan's mixing diagram. One can then apply Leray's spectral sequence of the fiber bundle MG → BG to compute the equivariant cohomology from the cohomology of M and the cohomology of the classifying space BG.