Computation of generalized equivariant cohomologies of Kac-Moody flag varieties

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.Comment: 19 pages, 6 figures, this is a new and completely modified version of DG/040207

The mod 2 cohomology of fixed point sets of anti-symplectic involutions

Let $M$ be a compact, connected symplectic manifold with a Hamiltonian action of a compact $n$-dimensional torus $G=T^n$. Suppose that $\sigma$ is an anti-symplectic involution compatible with the $G$-action. The real locus of $M$ is $X$, the fixed point set of $\sigma$. Duistermaat uses Morse theory to give a description of the ordinary cohomology of $X$ in terms of the cohomology of $M$. There is a residual \G=(\Zt)^n action on $X$, and we can use Duistermaat's result, as well as some general facts about equivariant cohomology, to prove an equivariant analogue to Duistermaat's theorem. In some cases, we can also extend theorems of Goresky-Kottwitz-MacPherson and Goldin-Holm to the real locus.Comment: 21 pages, 1 figur

The equivariant KK-theory and cobordism rings of divisive weighted projective spaces

We apply results of Harada, Holm and Henriques to prove that the Atiyah-Segal equivariant complex $K$-theory ring of a divisive weighted projective space (which is singular for nontrivial weights) is isomorphic to the ring of integral piecewise Laurent polynomials on the associated fan. Analogues of this description hold for other complex-oriented equivariant cohomology theories, as we confirm in the case of homotopical complex cobordism, which is the universal example. We also prove that the Borel versions of the equivariant $K$-theory and complex cobordism rings of more general singular toric varieties, namely those whose integral cohomology is concentrated in even dimensions, are isomorphic to rings of appropriate piecewise formal power series. Finally, we confirm the corresponding descriptions for any smooth, compact, projective toric variety, and rewrite them in a face ring context. In many cases our results agree with those of Vezzosi and Vistoli for algebraic $K$-theory, Anderson and Payne for operational $K$-theory, Krishna and Uma for algebraic cobordism, and Gonzalez and Karu for operational cobordism; as we proceed, we summarize the details of these coincidences.Comment: Accepted for publication in Tohoku Math.