Cohomology of GKM Fiber Bundles

TL;DR

This paper investigates the structure of equivariant cohomology in GKM fiber bundles, establishing a graph-theoretic Leray-Hirsch theorem and applying it to flag varieties, advancing understanding of GKM space cohomology.

Contribution

It introduces a graph-based version of the Leray-Hirsch theorem for GKM fiber bundles and applies it to the study of flag varieties' equivariant cohomology.

Findings

Established a graph-theoretic Leray-Hirsch theorem for GKM fiber bundles

Connected GKM graph cohomology to fiber bundle structures

Applied results to compute equivariant cohomology of flag varieties

Abstract

The equivariant cohomology ring of a GKM manifold is isomorphic to the cohomology ring of its GKM graph. In this paper we explore the implications of this fact for equivariant fiber bundles for which the total space and the base space are both GKM and derive a graph theoretical version of the Leray-Hirsch theorem. Then we apply this result to the equivariant cohomology theory of flag varieties.