Localization Formulas

This chapter highlights localization formulas. The equivariant localization formula for a torus action expresses the integral of an equivariantly closed form as a finite sum over the fixed point set. It was discovered independently by Atiyah and Bott on the one hand, and Berline and Vergne on the other, around 1982. The chapter describes the equivariant localization formula for a circle action and works out an application to the surface area of a sphere. It also explores some equivariant characteristic classes of a vector bundle. These include the equivariant Euler class, the equivariant Pontrjagin classes, and the equivariant Chern classes.