Any smooth map of compact oriented manifolds
induces a map in homology, which by Poincaré
duality, corresponds to a map in cohomology,
called the push-forward map or the Gysin map. The
calculation of the push-forward map for various
flag bundles plays an important role in
enumerative algebraic geometry.
Suppose a Lie group acts on a compact oriented
manifold with isolated fixed points.
The Atiyah--Bott--Berline--Vergne localization
formula converts an integral over the manifold to
a finite sum over the fixed points. Surprisingly,
this localization formula provides a systematic
method for calculating the Gysin map of a fiber
bundle, allowing us to recover some beautiful
classical formulas. This talk assumes no
knowledge of the localization formula. I will
explain the background in equivariant cohomology,
the localization formula, and how to apply it to
the problem of the Gysin map.