Balanced Fiber Bundles and GKM Theory

Abstract

Let T be a torus and B a compact T-manifold. Goresky et al. show in [3] that if B is (what was subsequently called) a GKM manifold, then there exists a simple combinatorial description of the equivariant cohomology ring H*[over]T(B) as a subring of H*[over]T(B[superscript 2]). In this paper, we prove an analog of this result for T-equivariant fiber bundles: we show that if M is a T-manifold and π:M→B a fiber bundle for which π intertwines the two T-actions, there is a simple combinatorial description of H*[over]T(M) as a subring of H*[over]T(π[superscript -1](B[superscript T])). Using this result, we obtain fiber bundle analogs of results of Guillemin et al. on GKM theory for homogeneous spaces.