T-equivariant cohomology of cell complexes and the case of infinite Grassmannians

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In 1998, Goresky, Kottwitz, and MacPherson showed that for certain spaces X equipped with a torus action, the T-equivariant cohomology ring of X can be described by combinatorial data obtained from its orbit decomposition. Thus, their theory transforms calculations of the equivariant topology of X to those of the combinatorics of the orbit decomposition. Since then, many authors have studied this interplay between topology and combinatorics. In this paper, we generalize the theorem of Goresky, Kottwitz, and MacPherson to the (possibly infinite-dimensional) setting where X is any equivariant cell complex with only even-dimensional cells and isolated T-fixed points, along with some additional technical hypotheses on the gluing maps. This generalization includes many new examples which have not yet been studied by GKM theory, including homogeneous spaces of a loop group LG.