Gysin formulas and equivariant cohomology

Abstract

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.