Computing the Gysin map using fixed points

TL;DR

This paper introduces a systematic method using equivariant cohomology localization to compute the Gysin map, simplifying calculations in algebraic geometry and extending classical formulas to more general fiber bundles.

Contribution

It demonstrates how localization in equivariant cohomology can be used to compute the Gysin map, providing a new approach to classical and generalized pushforward formulas.

Findings

Recovered classical pushforward formulas for flag bundles.

Extended formulas to fiber bundles with equivariantly formal fibers.

Provided a systematic localization-based computation method.

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.