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Search: MSC category 55N91 ( Equivariant homology and cohomology [See also 19L47] )

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1. CMB Online first

Crooks, Peter; Holden, Tyler
Generalized Equivariant Cohomology and Stratifications
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
Categories:55N91, 19L47

2. CMB 2014 (vol 58 pp. 80)

Harada, Megumi; Horiguchi, Tatsuya; Masuda, Mikiya
The Equivariant Cohomology Rings of Peterson Varieties in All Lie Types
Let $G$ be a complex semisimple linear algebraic group and let $Pet$ be the associated Peterson variety in the flag variety $G/B$. The main theorem of this note gives an efficient presentation of the equivariant cohomology ring $H^*_S(Pet)$ of the Peterson variety as a quotient of a polynomial ring by an ideal $J$ generated by quadratic polynomials, in the spirit of the Borel presentation of the cohomology of the flag variety. Here the group $S \cong \mathbb{C}^*$ is a certain subgroup of a maximal torus $T$ of $G$. Our description of the ideal $J$ uses the Cartan matrix and is uniform across Lie types. In our arguments we use the Monk formula and Giambelli formula for the equivariant cohomology rings of Peterson varieties for all Lie types, as obtained in the work of Drellich. Our result generalizes a previous theorem of Fukukawa-Harada-Masuda, which was only for Lie type $A$.

Keywords:equivariant cohomology, Peterson varieties, flag varieties, Monk formula, Giambelli formula
Categories:55N91, 14N15

3. CMB 2007 (vol 50 pp. 365)

Godinho, Leonor
Equivariant Cohomology of $S^{1}$-Actions on $4$-Manifolds
Let $M$ be a symplectic $4$-dimensional manifold equipped with a Hamiltonian circle action with isolated fixed points. We describe a method for computing its integral equivariant cohomology in terms of fixed point data. We give some examples of these computations.

Categories:53D20, 55N91, 57S15

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