Equivariant cohomology and equivariant intersection theory

This text is an introduction to equivariant cohomology, a classical tool for topological transformation groups, and to equivariant intersection theory, a much more recent topic initiated by D. Edidin and W. Graham. It is based on lectures given at Montréal is Summer 1997. Our main aim is to obtain explicit descriptions of cohomology or Chow rings of certain manifolds with group actions which arise in representation theory, e.g. homogeneous spaces and their compactifications. As another appplication of equivariant intersection theory, we obtain simple versions of criteria for smoothness or rational smoothness of Schubert varieties (due to Kumar [40], Carrell-Peterson [16] and Arabia [4]) whose statements and proofs become quite transparent in this framework. We now describe in more detail the contents of these notes; the prerequisites are notions of algebraic topology, compact Lie groups and linear algebraic groups. Sections 1 and 2 are concerned with actions of compact Lie groups on topological spaces, especially on symplectic manifolds. The material of Section 1 is classical: universal bundles, equivariant cohomology and its relation to usual cohomology, and the localization theorem for actions of compact tori. A useful refinement of the latter theorem is presented in Section 2, based on joint work with Michèle Vergne. It leads to a complete description of the cohomology ring of compact multiplicity-free spaces. Examples include coadjoint orbits, projective toric manifolds and De Concini-Procesi’s complete symmetric varieties [18]. The subject of the last three sections is equivariant intersection theory for actions of linear algebraic groups on schemes. Edidin and Graham’s equiv-