Mathematics – Algebraic Topology
Scientific paper
2004-09-17
Mathematics
Algebraic Topology
19 pages, 6 figures, this is a new and completely modified version of DG/0402079
Scientific paper
In 1998, Goresky, Kottwitz, and MacPherson showed that for certain projective varieties X equipped with an algebraic action of a complex torus T, the equivariant cohomology ring H_T(X) can be described by combinatorial data obtained from its orbit decomposition. In this paper, we generalize their theorem in three different ways. First, our group G need not be a torus. Second, our space X is an equivariant stratified space, along with some additional hypotheses on the attaching maps. Third, and most important, we allow for generalized equivariant cohomology theories E_G^* instead of H_T^*. For these spaces, we give a combinatorial description of E_G(X) as a subring of \prod E_G(F_i), where the F_i are certain invariant subspaces of X. Our main examples are the flag varieties G/P of Kac-Moody groups G, with the action of the torus of G. In this context, the F_i are the T-fixed points and E_G^* is a T-equivariant complex oriented cohomology theory, such as H_T^*, K_T^* or MU_T^*. We detail several explicit examples.
Harada Megumi
Henriques Andre
Holm Tara S.
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