An introduction to equivariant cohomology and the equivariant first Chern class

The goal of this paper to provide a relatively accessible and integrated introduction to the algebraic topology of spaces with Lie group actions, in both the smooth and the holomorphic category. We present a detailed treatment of the basic constructions in equivariant de Rham theory and Dolbeault theory. We also discuss equivariant connections and curvature on vector bundles equipped with infinitesimal lifts, as well as the equivariant first Chern class of line bundles with infinitesimal lifts. A novel feature of this presentation is the definition of infinitesimal lift via differential operators on the vector bundle.