  Under Poincar\'e duality, a smooth map of compact oriented manifolds induces
a pushforward map in cohomology, called the "Gysin map." It plays an important
role in enumerative geometry. Using the equivariant localization formula, the
author gave in 2017 a general formula for the Gysin map of a fiber bundle with
equivariantly formal fibers. Equivariantly formal manifolds include all
manifolds with cohomology in only even degrees such as complex projective
spaces, Grassmannians, and flag manifolds as well as $G/H$, where $G$ is a
compact Lie group and $H$ is a closed subgroup of maximal rank. This article is
a simplified exposition using the example of a projective bundle to illustrate
the algorithm.
