  We give two generalizations of the Atiyah-Bott-Berline-Vergne localization
theorem for the equivariant cohomology of a torus action: 1) replacing the
torus action by a compact connected Lie group action, 2) replacing the manifold
having a torus action by an equivariant map. 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.
