Balanced fiber bundles and GKM theory

TL;DR

This paper extends GKM theory to T-equivariant fiber bundles, providing a combinatorial description of their equivariant cohomology rings, and applies this to homogeneous spaces.

Contribution

It generalizes GKM theory from GKM manifolds to T-equivariant fiber bundles, offering new combinatorial tools for their cohomology.

Findings

Provides a combinatorial description of H_T^*(M) for fiber bundles

Extends GKM theory to a broader class of T-manifolds

Derives results for homogeneous spaces using the new framework

Abstract

Let $T$ be a torus and $B$ a compact $T-$manifold. Goresky, Kottwitz, and MacPherson show in \cite{GKM} that if $B$ is (what was subsequently called) a GKM manifold, then there exists a simple combinatorial description of the equivariant cohomology ring $H_T^*(B)$ as a subring of $H_T^*(B^T)$. In this paper we prove an analogue of this result for $T-$equivariant fiber bundles: we show that if $M$ is a $T-$manifold and $\pi \colon M \to B$ a fiber bundle for which $\pi$ intertwines the two $T-$actions, there is a simple combinatorial description of $H_T^*(M)$ as a subring of $H_T^*(\pi^{-1}(B^T))$. Using this result we obtain fiber bundle analogues of results of \cite{GHZ} on GKM theory for homogeneous spaces.