Generalized equivariant cohomology and stratifications (Q2827401)

Let \(T\) be a compact torus and let \(T_{\mathbb C}\) denote its complexification. Let \(E^\ast_{T}\) be a generalized \(T\)-equivariant cohomology. In this paper the authors provide a systematic framework to compute \(E^\ast_{T}(X)\) for a smooth complex projective \(T_{\mathbb C}\)-variety \(X\) with an equivariant stratification. Specifically this research is focused on the case where \(E^\ast_{T}\) is ordinary equivariant cohomology \(H^\ast_{T}\), equivariant \(K\)-theory \(K^\ast_{T}\) or equivariant complex cobordism \({MU}^\ast_{T}\). Indeed, in this case the authors prove that if \(X\) can be written as the direct limit of a sequence \(X_0\subseteq X_1\subseteq X_2\subseteq\cdots\) of equivariant closed embeddings of smooth complex projective \(T_{\mathbb C}\)-varieties, each \(X_i\) having finitely many \(T\)-fixed points, then there is an isomorphism NEWLINE\[NEWLINEE^\ast_{T}(X)\cong\prod_{x\in X}^{T} E^\ast_{T}({pt})NEWLINE\]NEWLINE of \(E^\ast_{T}({pt})\)-modules (Theorem 1). In addition, the authors present an example of a \(T\)-space \(X\) satisfying the hypotheses posed above. Let \(G\) be a simply-connected complex semisimple Lie group and \(T\subseteq G\) be a maximal torus. If we denote by Gr its affine Grassmannian, then we find that Gr actually is the desired \(T\)-space having a natural stratification. In fact, by using the ideas discussed for proving the above isomorphism the authors obtain \(E^\ast_{T}(\mathrm{Gr})\cong \prod_{\lambda\in X_\ast(T)} E^\ast_{T}({pt})\) as \(E^\ast_{T}({pt})\)-modules. Here \(X_\ast(T)=\mathrm{Hom}(\mathbb C^\ast,T_{\mathbb C})\).