Equivariant formality and localization formulas.

摘要

We present a brief introduction to equivariant cohomology. Particular emphasis is made to the Cartan model for equivariant cohomology. The Cartan model is a differential complex that computes the equivariant cohomology of a G-manifold M, where G a compact connected Lie group. Throughout this thesis we work in the category of G-manifolds and smooth G-maps.; Using the Cartan model to compute the equivariant cohomology, H*G (M), of a G-manifold M, we give a description of the image of the edge morphism j*:H*G (M) → H*(M). Based on this description, we give an equivalent characterization of an equivariantly formal G-manifold M, in terms of the H*(BG)-module structure of H*G (M).; Finally, we present a generalization of the Atiyah-Bott-Berline-Vergne localization formula. Specifically, given a T-map f : M → N between compact oriented T-manifolds we give a formula for the push-forward map in equivariant cohomology, (fT)* : H*T (M) → H*(N), in terms of data from the fixed-point set of M and N. As an application of the relative localization formula, we compute the equivariant push-forward for the inclusion map mu : O → G, of a conjugacy class O of G. The map mu : O → G corresponds to the Lie-group-value moment map for O, introduced by A. Alekseev, A. Malkin and E. Meinrenken in [1].