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# Generalized Equivariant Cohomology and Stratifications

Published:2016-06-22
Printed: Sep 2016
• Peter Crooks,
Department of Mathematics, University of Toronto , Toronto, ON M5S 2E4
• Tyler Holden,
Department of Mathematics, University of Toronto , Toronto, ON M5S 2E4
 Format: LaTeX MathJax PDF

## Abstract

For $T$ a compact torus and $E_T^*$ a generalized $T$-equivariant cohomology theory, we provide a systematic framework for computing $E_T^*$ in the context of equivariantly stratified smooth complex projective varieties. This allows us to explicitly compute $E_T^*(X)$ as an $E_T^*(\text{pt})$-module when $X$ is a direct limit of smooth complex projective $T_{\mathbb{C}}$-varieties with finitely many $T$-fixed points and $E_T^*$ is one of $H_T^*(\cdot;\mathbb{Z})$, $K_T^*$, and $MU_T^*$. We perform this computation on the affine Grassmannian of a complex semisimple group.
 Keywords: equivariant cohomology theory, stratification, affine Grassmannian
 MSC Classifications: 55N91 - Equivariant homology and cohomology [See also 19L47] 19L47 - Equivariant $K$-theory [See also 55N91, 55P91, 55Q91, 55R91, 55S91]

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