The Weil Model and the Cartan Model

Abstract

The results of the last chapter suggest that, for any G⋆ module B we take B⊗ W as an algebraic model for the X × E of Chapter 1, and hence H _bas (B ⊗ W) as a definition of the equivariant cohomology of B. In fact, one of the purposes of this chapter will be to justify this definition. However the computation of (B ⊗ W)_bas is complicated. So we will begin with a theorem of Mathai and Quillen which shows how to find an automorphism of B ⊗ W which simplifies this computation. For technical reasons we will work with W ⊗ B instead of B ⊗ W and replace W by an arbitrary W⋆ module.