On the localization formula in equivariant cohomology

We give a generalization of the Atiyah–Bott–Berline–Vergne localization theorem for the equivariant cohomology of a torus action. We replace the manifold having a torus action by an equivariant map of manifolds having a compact connected Lie group action. This provides a systematic method for calculating the Gysin homomorphism in ordinary cohomology of an equivariant map. As an example, we recover a formula of Akyildiz–Carrell for the Gysin homomorphism of flag manifolds.