Balanced fiber bundles and GKM theory

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 $H_T^{\ast }(B)$ as a subring of $H_T^{\ast }(B^T)$. 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 $\pi :M\to B$ a fiber bundle for which $\pi$ intertwines the two $T-$actions, there is a simple combinatorial description of $H_T^{\ast }(M)$ as a subring of $H_T^{\ast }({\pi }^{-1}(B^T))$. Using this result we obtain fiber bundle analogues of results of \cite{GHZ} on GKM theory for homogeneous spaces.

14 pages

References in corpus (1)

• Cohomology of GKM Fiber Bundles