The equivariant-cohomology tag has no usage guidance.

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### Reference request: Equivariant Topology

I am teaching a graduate seminar in equivariant topology. The format of the course is that I will give 2-3 weeks of background lectures, then each week a student will present a topic. The students ...

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### $RO(G)$-graded homotopy groups vs. Mackey functors

Everything here is model-independent: either take co/fibrant replacements wherever appropriate, or work $\infty$-categorically.
Also, I've looked through other similar MO questions, but I didn't find ...

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### How is an $S^1$-equivariant elliptic cohomology theory affected as we continuously vary the underlying elliptic curve?

Grojnowski constructs a $S^1$-equivariant cohomology theory over a complex elliptic curve $E$, designed to trivially satisfy: $$E^*_{S^1}(pt) = E$$
The functor $E^*_{S^1}(-)$ takes in a space $X$ ...

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### Microlocalizing Hochschild homology

A recent paper of Ben-Zvi and Nadler gives a general formalism for understanding "dimensions" in sheaf theories. Without getting too far into details, amongst other things, this formalism allows us ...

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### Passing from T-equivariant to G-equivariant cohomology

Let G=GLn(ℂ) and let T be a maximal torus. Let X be a topological space with a G-action. My question is: when is the canonical map $$H^*_G(X;\mathbb{Z})\to H^*_T(X;\mathbb{Z})$$ injective?
Some ...

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### T-equivariant cohomology of flag variety

Let $X=G/B$ , where $G=GL_n(\mathbb{C}^n)$ and $B$ be the upper triangular matrices. I am curoius about the structure of $H^*_T(G/B)$ which I consider as a $H_T^*(pt)$-module. If we just consider ...

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### Equivariant motivic sheaves

Thanks to the work of Cisinski-Deglise: http://arxiv.org/abs/0912.2110, we now have a triangulated category of `motivic sheaves' available that admits the standard yoga of the six functors.
Is there ...

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### Equivariant and orbifold Chern classes

Edit. After thinking about this problem a bit longer, I am not so sure anymore that the Bredon cohomology proposed by Adem and Ruan gives me the invariants I am looking for. I have therefore moved ...

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### Is there any “deep” relation between the localization theorem of equivariant cohomology and the localization theorem of equivariant K-theory

First let's consider equivariant cohomology: if a compact Lie group $G$ acts on a compact manifold $M$. We have the equivariant cohomology $ H_G(M)$ defined as the cohomology of the cochain complex $((...

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### Finite generation of equivariant cohomology rings

Let $G$ be a finite group acting on a topological space $X$. (In my applications, $X$ is the classifying space of a compact Lie group.) Let $H^\ast(-)=H^\ast(-;\mathbb{Z}/2\mathbb{Z})$ denote mod 2 ...

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### How does the kernel of the map $\Omega^{\bullet}(X)\rightarrow \Omega^{\bullet}(G\times X)$ relate to equivariant cohomology?

This question may be trivial for experts. Consider a (compact, connected) smooth manifold $X$ and a (compact connected) Lie group $G$ act on $X$. Then we have the action map
$$
\mu: G\times X\...

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### Citation: earliest incidence of the Borel localization theorem

The Borel localization theorem in (Borel) equivariant cohomology states that if $T$ is a torus and $M$ a smooth $T$-manifold, with fixed point set $M^T$, then upon localizing the coefficient ring $H^*...

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### Equivariant Stratifications of a Variety

Let $X$ be a complex variety acted upon algebraically by a complex torus $T$. Suppose that $\{X_{\beta}\}_{\beta\in S}$ is a finite $T$-equivariant stratification of $X$, so that the $X_{\beta}$ are ...

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### Twisted equivariant modular forms?

I'd like to know where I can find information about a class of objects which I think deserve to be called twisted equivariant modular forms. Let me guess a definition, indicate how it can be made more ...

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### Rational homology and finite group actions

I'm looking for examples of the following phenomena. Let $X$ be a reasonable space (say, a CW complex) and $G$ be a finite group acting on $X$. For all $k \geq 1$, the projection map $X \rightarrow ...

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### Change of groups for naive G-spectra

Let $H$ be a subgroup of $G$ a compact Lie group and
let $\text{spectra}[G]$ be the category of naive $G$-spectra (ie G-objects in the category of spectra). Then there is a forgetful functor $i^*$ ...

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### A Property of Generalized Equivariant Cohomology

Let $G_i$ be a compact Lie group, $i=1,2$, and let $E_{G_i}^*$ be a $\mathbb{Z}$-graded complex-oriented $G_i$-equivariant generalized cohomology theory with commutative products. Let $X_i$ be a ...

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### Example request: equivariant formality versus formality for homogeneous spaces

Recall that a continuous action of a compact Lie group $T$ on a space $X$ is said to be equivariantly formal if the Borel equivariant cohomology $H^*(X \times_T ET;\mathbb Q)$ surjects through ...

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### Mackey(also Green and Tambara) functors and Greenlees-May

This is somewhat related to a question that I asked on Math.SE but, sadly, received no response. I apologize ahead of time if this is not appropriate for MO. Feel free to vote to close if this is the ...

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### Geometric construcion of Proj as a quotient by a $\mathbb{G}_m$ action

I'm trying to translate the Proj construction as a kind of quotient by a $\mathbb{G}_m$ action. Here's what I have so far:
Let $X=Spec A$ be an affine scheme (after this case is setteled I imagine it ...

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### $RO(G)$-Graded Cohomology Theories

Let $G$ be a compact Lie group with real representation ring $RO(G)$. Recently, I have been learning about some aspects of $RO(G)$-graded cohomology theories (for a precise definition, see Chapter ...

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### Is a Lie group equivariantly formal under conjugation by a maximal torus?

Given an action of a group $G$ on a topological space $X$, the associated homotopy quotient is $$X_G := (EG \times X)/G,$$ where $EG$ is the total space of a universal principal $G$-bundle $EG \to BG$ ...

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### Is there a Whitehead-type theorem in T-equivariant cohomology?

Let $T$ be a real torus, and let $X$ and $Y$ be $T$-spaces. Under what conditions (if any) will the existence of graded $H^*_T$-algebra isomorphism between the $T$-equivariant cohomologies of $X$ and $...

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### Borel constructions, equivariant cohomology, and homotopy quotients of monoid actions.

Let $M$ be a (discrete) monoid acting on a space $X.$ We may take the quotient of $X,$ by this action, $X/M$ that is the coequalizer of the action map $M \times X \to X$ against the projection map. $X$...

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### Künneth formula for Bredon cohomology theory

Let $G$ be a finite group. Let $X$ and $Y$ be two $G$-CW complexes with known integer graded $G$-equivariant Bredon cohomology with constant coefficient systems. Is there any Künneth formula for this ...

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### Localization in equivariant cohomology theory for groups other than ($p$-)tori

Recall the following localization theorem, as stated in Hsiang's Cohomology Theory of Compact Transformation Groups:
Theorem. Let $G=(S^1)^k$ be a torus, $X$ a paracompact $G$-space with finite ...

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### Relation of BRST model of equivariant cohomology and BRST cohomology?

I'm now reading Kalkman's paper "BRST model for equivariant cohomology and representation for equivariant Thom class". And I've seen his definition for BRST model is
$B=W(\mathfrak{g})\otimes \...

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### Computing the Product Structure in Equivariant Cohomology via a Stratification and Thom-Gysin

Let $X$ be a smooth complex projective variety acted upon algebraically by a complex torus $T$. Suppose that $\{X_{\beta}\}_{\beta\in B}$ is a finite $T$-invariant stratification of $X$ into smooth ...

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### extension of cohomology theories

In Rudyak's book it is proved (3.22 page 160) that every additive cohomology theory (in the sense of Eilenberg-Steenrod) defined on the category of finite dimensional pointed CW complexes can be ...

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### Equivariant Cohomology for actions with finite stabilizers

Let $X$ be a reasonable topological space (let's say it has the homotopy type of a CW complex) and let $G$ be a topological group acting on that space. Let $E_G \rightarrow B_G$ be the universal ...

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### Advantage in Using Cyclic Homology to a compute Equivariant (Co)Homology of Loop Spaces

I am trying to compute equivariant (co)homology of the free loop space of a manifold $M$ that is not a Lie group, $H^{S^1}_*(LM)$ with the natural rotation action of $S^1$ on the loops of the free ...

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### $GL_k$-equivariant cohomology of $k\times n$ matrices

I'm having a surprisingly hard time finding references for some facts about $GL_k$-equivariant cohomology of the space of $k\times n$ matrices. Specifically, I believe the following things to be true:
...

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### Equivariant cohomology of finite group actions and invariant cohomology classes

Let $W$ be a finite group acting on a space $X$. In what generality is it true that $H^*_W(X) = H^* (X)^W$? We always have a map $H^*_W(X) \rightarrow H^* (X)^W$, but it is certainly not an ...

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### Explicit computation of Gromov-WItten invariants

After studying some foundation of Gromow-Witten invariants, I now would like to see an explicit computation. I heard that one should first take a look at the total space of $\mathcal{O}(-1)^{\oplus2}$ ...

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### Equivariant cohomology and complex non-degenerate bilinear forms

Let $M = GL(n,\mathbb{C})$ be the set of non-degenerate bilinear forms on $\mathbb{C}^n$ (not necessarily symmetric). The general linear group $G = GL(n,\mathbb{C})$ acts on $M$ in the usual way
$$G\...

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### Using and understanding the Atiyah-Bott localization theorem/integration formula

I posted this on r/math, but was told I might have better success here given the level of the question.
Basically, I need to learn how to use the localization theorem to compute integrals on ...

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### Equivariant Poincare Series of Based Loop Group of SU(2)

Let $\Omega SU(2)$ denote the based loop group of $SU(2)$, and consider the action of $S^1$ on $\Omega SU(2)$ as a maximal torus of $SU(2)$. (This is not the "loop rotation" action.) Is there an ...

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### Why is the equivariant Euler class a character ?

Let us first precise the question : let $T$ be a torus, $\alpha : T \to \mathbb{C}$ be an irreducible character. I am interested in the $T$-equivariant Euler class of the ($T$-equivariant) bundle $\...

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### Formality of classifying spaces (for not necessarily connected groups)

As should be evident from the title this question has a similar flavor to:
Formality of classifying spaces
However, unlike Geordie's question, I will be working with torsion free coefficients (say ...

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### Example equivariant Mayer-Vietoris for $H^{*}_{S^{1}}(S^{2})$

We want to calculate $H^{k}_{S^1}(S^2)$. We can choose two open sets $U=S^{2} \setminus p_{+}$ and $V = S^{2} \setminus p_{-}$, where $p_{+}$ and $p_{-}$ are the north and south pole of $S^{2}$. They ...

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### Moment map of equivariant line bundles

I'm reading Szabo's `Equivariant Cohomology and Localization of Path Integrals'. I've stumbled upon an equation I can't make sense of, in the discussion about $G$-equivariant line bundles on ...

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### Calculation of RO(G)-graded cohomology of a point

Recently I have been learning about some calculation of RO(G)-graded cohomology of a point from J.L. Caruso's paper on "Operations in equivariant Z/p-cohomology".
My question is how to calculate RO(G)...

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### J-function of cotangent bundle of complete flag variety

Givental and Kim showed that the $J$-function of the complete flag variety $Fl_n=SL_{n}/B$ becomes an eigenfunction of the Toda Hamiltonian. How about the $J$-function of the cotangent bundle $T^*...

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### Equivariant version of cyclic versus de Rham (co)homology of commutative algebras?

Let $A$ be a commutative algebra that is a $g$-module, for some Abelian Lie algebra $g$. The primary example of my interest is when $A$ is the ring of functions over an affine variety, say $\mathbb{C}^...

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### Equivariant Cohomology of Non-Compact Spaces via Fixed Points

Let $T$ be a complex torus and $X$ a smooth quasi-projective $T$-variety with finitely many fixed points. Denote by $\varphi:H_{T}(X)\rightarrow H_{T}(X^T)$ the map on equivariant cohomology induced ...

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### Analogue of Borel--Bott--Weil for General Equivariant Vector Bundles

The Borel--Bott--Weil Theorem gives the dimensions of the cohomology groups of the equivariant line bundles over flag manifolds. Does there exist an analogous result for general equivariant vector ...

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### cohomology of the orbit space of a group action

Let $M$ be a manifold. Let a finite group $G$ act on $M$ discretely. Let $F$ be a field.
Suppose the induced action of $G$ on the cohomology algebra $H^*(M,F)$ is known. We want to obtain $H^*(M/G;F)$...

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### Simple Equivariant homology [no borel-Moore]

Hey. I'm working with Bredon's equivariant cohomology. At some point I need to compute the $4$th equivariant cohomology group of $S^1 \x D^3$ relatively to its boundary for the antipodal action of $\...

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### $G$-CW complex structure of universal a $\mathcal{F}$-space

Let $G$ be a finite group and $H$ be an abelian subgroup of $G$. Let $\mathcal{F}$ be a family of all subgroups of $H$ , i.e. $\mathcal{F}= \{K : K \leq H\}$ Define universal $\mathcal{F}$-space $E\...

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### Is the equivariant Gysin map an $H_G^*(\text{pt})$-module morphism?

Let $G$ be a complex reductive group, $X$ a smooth projective variety on which $G$ acts algebraically, and $Y \subseteq X$ a $G$-invariant smooth closed subvariety such that $X\setminus Y$ is also ...