Everytopic Seminar

 October 24th, 2008, 1:40 pm – 3:00pm, Goldsmith 226



Title: Computing the Gysin homomorphism using fixed points

Speaker: Loring Tu, Tufts University



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.