Computing the Gysin map using fixed points

Abstract

The Gysin map of a map between compact oriented manifolds is the map in cohomology induced by the push-forward map in homology. In enumerative algebraic geometry, formulas for the Gysin map of a flag bundle play a vital role. These formulas are usually proven by algebraic or combinatorial means. This article shows how the localization formula in equivariant cohomology provides a systematic method for calculating the Gysin homomorphism in the ordinary cohomology of a fiber bundle. As examples, we recover classical pushforward formulas for generalized flag bundles. Our method extends the classical formulas to fiber bundles with equivariantly formal fibers.