Questions tagged [equivariant-cohomology]

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3
votes
0answers
191 views

What is rigidity of Hirzebruch, and Witten genera?

I would like to find some good references (or any insight) that would help me to understand a few articles mentioning rigidity of Hirzeburch genus. One of the consequences of this phenomenon is that ...
3
votes
0answers
109 views

Computations of Bredon homology of $S(1+\sigma)$ with Universal Coefficient S.S

What I am trying to do is to compute $\mathbb{Z}$-graded Bredon homology of $S(1+\sigma)$ over $Q\times\Sigma_2$, where $Q$ is a cyclic group of order 2 $\sigma$ is its real sign representation $\...
2
votes
1answer
149 views

Equivariant Coefficient ring action on singular cohomology

Let $X$ be a manifold acted on by a Lie group $G$. The $G$-equivariant cohomology of $X$ with coefficients in a ring $\mathcal{R}$ is defined as the cohomology ring $$ H_G^*(X; \mathcal{R}) := H^*(X_G;...
10
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4answers
948 views

Reading list for Equivariant Cohomology

I was applying equivaraint cohomology, in particular in the symplectic setting, for some time, but I feel like I am missing some nice books/course notes/articles. Could you advise me some literature, ...
5
votes
1answer
228 views

Equivariant cohomology algebra of toric variety

Let $X$ be a complex projective and smooth toric variety of complex dimension $n$. It is acted by the real torus $T=(S^1)^n$. Is it true that the $T$-equivariant cohomology $H^*_T(X,\mathbb{Z})$ ...
4
votes
1answer
137 views

Transition equations for double Schubert polynomials

For $w$ a permutation, the associated (ordinary) Schubert polynomial $\mathfrak{S}_w(\textbf{x})$ is a multivariate polynomial that represents for the cohomology class of the Schubert variety $X_w$ in ...
2
votes
1answer
195 views

A question on relative equivariant cohomology

Suppose that we have defined an extraordinary $G$-equivariant cohomology theory $H$ (say $G$ is a compact group). If $X$ is a $G$-space and $A\subset X$ is a closed $G$-equivarant contractible ...
2
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0answers
85 views

Computation of equivariant 3 form

I want to how an equivariant 2-form and equivariant 3- form look like i,e., Let M be a complex Manifold say $ S^4 \subset C^3$ and a compact lie group $S^1$ acting on it via the action $\exp{i\theta}....
7
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1answer
259 views

Differentials in Weil model for equivariant cohomology

Why should we define the differential in Weil model as follows? I could understand $\sum_{j,k} c_{jk}^i \theta_j \wedge \theta_k$ plays a role in the formula because it is the dual of the structure ...
12
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1answer
1k views

The homology of the orbit space

Suppose we have an acyclic group $G$ and let $X$ be a contractible CW-complex such that $G$ acts freely on $X$ (we do not suppose that the action is proper). Is there a way to understand the homology ...
5
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0answers
127 views

Equivariant L-infinity structure associated to a DGBV algebra

Let $V$ be a differential graded Batalin-Vilkovisky (DGBV) algebra over say $\mathbb{C}$. This means that $V$ is a commutative differential graded algebra (CDGA), with differential $\partial$ of ...
4
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0answers
68 views

Formality of fixed points in the equivariant localisation

Let $X$ be a complex algebraic variety equipped with an algebraic $\mathbf{C}^{\times}$-action. The Borel construction gives a map $f: \mathrm{E}\mathbf{C}^{\times}\times^{\mathbf{C}^{\times}}X\to \...
3
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1answer
179 views

Relative equivariant cohomology

Let us assume that $X=\mathbb{R}\times S^1$ is given with a $G=\mathbb{Z}_2$ action that corresponds to the symmetry $(x,e^{i\theta})\mapsto(-x,e^{-i\theta})$. I want to compute the equivariant ...
6
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0answers
166 views

A Poincare dual spectral sequence for equivariant cohomology: extension to generalized cohomology?

This question is a follow-up to my previous question: "Rotated" version of the Atiyah-Hirzebruch spectral sequence In that question, I discussed two different spectral sequences for ...
11
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1answer
331 views

Is there a kind of Poincare duality for Borel equivariant cohomology?

Let $G$ be a finite (or discrete) group, $M$ a $d$-dimensional manifold with smooth $G$-action (I am interested in the case where the action is not free, so $M/G$ is not a manifold). For an Abelian ...
7
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0answers
278 views

Equivariant De Rham theorem for orbifolds

Recall that the "classical" equivariant De Rham theorem states that, for a compact Lie group $G$ acting on a compact smooth manifold $M$, $$H_G^*(M,\mathbb{R})\cong H^*(\Omega_G(M)),$$ where $H_G^*(M,\...
5
votes
1answer
185 views

Transgression image and Serre spectral sequence for tori

Let $\mathbb{K} \subset \mathbb{T}$ be two tori acting on a topological space $X$ (with all the properties you want). We use the notations $$X_{\mathbb{T}} := (X \times E \mathbb{T}) / \mathbb{T}, \...
2
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1answer
183 views

Do Mackey (co)homology functors factor through derived categories? References with details?

Let $G$ be a compact Lie group; I will write $SH_G$ for the (equivariant) stable homotopy category of $G$-spectra (say, with respect to a complete universe; does its choice affect the homotopy ...
0
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1answer
109 views

Compute the rational cohomology ring $H^*(M;\mathbb{Q})$ of a product of Lie groups (and their quotients)

As the following product is a bit unfamiliar to me: How do we compute the rational cohomology ring $H^*(M;\mathbb{Q})$ of the product of Lie groups: $M=SO(n_1)\times U(n_2)\times SU(n_3)\times (...
2
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0answers
113 views

Reference request: explicit equivariant localization formula on toric complete intersections

This post is about an equivariant integration formula in a famous paper https://arxiv.org/pdf/alg-geom/9701016.pdf by Alexander Givental, where the author presented the formula without proof or ...
16
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1answer
571 views

“Rotated” version of the Atiyah-Hirzebruch spectral sequence

Let $G$ be a group, $X$ a topological space with $G$-action. For an Abelian group $A$, let $\mathcal{C}^n(X,A)$ be the group of $n$-cochains on $X$ with $A$ coefficients. We can treat this as a $G$-...
2
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0answers
106 views

Homotopy orbits and Local Cohomology Theorem

I'm procedding with understanding Greenlees's paper "The Four Approaches to Cohomology Theories with Reality": https://arxiv.org/abs/1705.09365 Problem concerns the section 1.D. Consider the cofiber ...
4
votes
1answer
341 views

Computing the Euler class of a vector bundle

I'm having the following problem: let $T \subset G := SO(2k)$ be the maximal torus acting on $V := \mathbb{R}^{2k}$ by linear transformations on each $2$-dimensional component. Denote by $V_T := (V \...
2
votes
2answers
157 views

Bredon cohomology of a sign representation for a cyclic group of order 4

Yet another question "I compute Bredon cohomology of something and I am not sure, whether it is correct". So I am taking a sign representation $\sigma$ of cyclic group of order 4, $C_4$. Then I ...
8
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1answer
238 views

Equivariant bundles invisible in K-theory and Borel cohomology

For a given topological group $G$ there are natural transformations $$K^* \leftarrow K^*_G \overset a\to H^{**}(EG \times_G -;\mathbb Q)$$ from equivariant K-theory, the first forgetting the $G$-...
3
votes
0answers
136 views

Finding generators of equivariant cohomology

Let $(M,\omega)$ be a symplectic manifold with symplectic form $\omega$, carrying a Hamiltonian action of a compact connected Lie group $G$ with moment map $\mu:M\to \mathfrak{g}^\ast$, where $\...
10
votes
0answers
232 views

Is there a spectral sequence of Atiyah's topological KR-theory that can be used to compute basic examples?

For Segal's complex $G$-equivariant $K$-theory, it is well-known that there is an Atiyah-Hirzebruch spectral sequence. If say $G$ a finite group and $X$ a finite CW-complex, the second page of this ...
9
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0answers
154 views

Fixed-points of a topological circle action

Suppose the circle group $G = S^1$ acts on $X$. If $X$ is a closed smooth manifold (and the action is smooth), then we know the fixed-points $X^G$ are a disjoint union of smooth submanifolds of $X$. ...
6
votes
0answers
220 views

Geometric interpretation of a formula for the induced character (fix point localization?)

Let $H < G$ be a subgroup of a finite group $G$. Let $X:=G/H$ and $\mathcal{F} \in Sh_G(X)$ be an equivariant sheaf on $X$ (w.r.t. left multiplication) associated to a finite dimensional ...
18
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2answers
529 views

Allowing $G$-CW complexes to have more general cells

Let $G$ a finite group. I've seen three options discussed for making $G$-cell complexes: in increasing generality, one might allow $X_n$ to be constructed from $X_{n-1}$ by attaching cells of the form ...
2
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0answers
205 views

Localization of the pushforward in equivariant cohomology

I am reading Nekrasov's paper and in page 2 he considers the $G \times T^2$ equivariant cohomology of the (compactified) moduli space $\tilde{M_k}$ of $U(N)$ instantons on $\mathbb{C}^2$. Here $G$ ...
2
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0answers
69 views

On the preservation of equivariant cohomology in certain quotients

Let $G$ be a discrete, countable group and $X$ a finite, free, proper $G$-CW complex, such that the underlying $CW$-structure of $X$ is locally finite. Further, let $C_*(X,G)$ be the induced free-...
7
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1answer
647 views

Homotopy theoretic description of homotopy fixed points (and obstructions) for an action of group $G$ on a groupoid $X$

There are several scattered statements about fixed points and obstructions which I'd very much like to see unified in some framework. To state them let $G$ be a group acting on a connected (1-...
6
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0answers
127 views

Binary forms and equivariant derived category

One of the classical questions in invariant theory is the classification of binary forms, i.e., the description of polynomial invariants of the ${\rm SL}_2(\mathbb{C})$-action on ${\rm Sym}^d \mathbb{...
5
votes
1answer
196 views

Equivariant cohomology ring is an integer domain

Let $G$ be a connected compact Lie group and let $V$ be a complex $G$-representation. Denote by $\mathbb{P}(V)$ the projectivization of the vector space $V$. I would like to ask a couple of questions ...
10
votes
2answers
488 views

Relation between affine flag and Grassmannian Steinberg variety

Let $\mathcal{K}=\mathbb{C}((t))$ be the field of formal Laurent series over $\mathbb{C}$, and by $\mathcal{O}=\mathbb{C}[[t]]$ the ring of formal power series over $\mathbb{C}$. Given a semi-simple ...
7
votes
1answer
451 views

Naive equivariant transfer

Given a group $G$, a $\mathbb Z$-graded cohomology theory $E^*_G$, and a $n$-sheeted covering $p\colon X \to Y$, I would like a transfer map $$p_!\colon E^*_G X \to E^*_G Y$$ satisfying $$\require{...
2
votes
1answer
103 views

Orbit decomposition of the restriction of an equivariant sheaf?

All sets and groups in the question are finite. In order to understand equivariant sheaves better I'm trying to prove some basic facts from Mackey theory using equivariant sheaves. The main obstacle ...
3
votes
1answer
165 views

Show the Cartan 3-form transgresses to the Killing form in the Weil algebra

Let $G$ be a connected, reductive Lie group, and $W\mathfrak g = (S[\mathfrak g^\vee] \otimes \Lambda[\mathfrak g^\vee],\delta)$ the associated Weil algebra. This is a CDGA equipped with an action of $...
4
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1answer
199 views

Functors between categories of equivariant sheaves are equivariant sheaves on the product?

This is a follow up question to this question which remained unanswered (satisfactorily) even after a large bounty. I have made a litlle progress and I have no a more specific question which might be ...
7
votes
1answer
406 views

Computing the equivariant cohomology of a specific $(\mathbb{Z}/2\mathbb{Z})^2$-space

In the paper On the Castelnuovo-Mumford regularity of the cohomology ring of a group, Symonds describes the following space. Let $G = (\mathbb{Z}/2\mathbb{Z})^2 = \{1,a,b,ab\}$ be an elementary ...
8
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1answer
429 views

Interactions (functors) between equivariant sheaves for different groups?

Let $G$ be a finite group and $k$ a field (alg. closed char 0 for simplicity). To every $G$ set $X$ we can assign the category of $G$-equivariant sheaves of $k$-vector spaces $Sh_G(X)$. It is ...
2
votes
2answers
283 views

How is the equivariant cohomology of a space related to the cohomology of the corresponding associated bundle

Let $X$ be a manifold with a left $G$-action, and let $\Sigma$ be a Riemann surface. How is the equivariant cohomology $H^*_G(X)$ of $X$ related to the de Rham cohomology of the associated bundle $H^*(...
3
votes
1answer
238 views

Equivariant characteristic classes on $\mathbb{P}^n$

Let $T=(\mathbb{C}^*)^n$ act on $\mathbb{P}^n$ torically by $$t.[x_0:\dots:x_n]=[x_0\;:\;t_1x_1\;:\;\ldots \;:\;t_nx_n]$$ I would like to know an expression for the equivariant Chern character $\...
9
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0answers
290 views

Equivariant obstruction theory done wrong

Let $G$ be a finite group, and let $p:E\to B$ be a $G$-fibration with fibre $F$. The correct framework for studying equivariant sections of $p$ is Bredon cohomology. With the right definitions, most ...
15
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2answers
1k views

Equivariant cohomology vs. invariant cohomology vs. cohomology of quotient space

Given a space $X$ and an action of a group $G$ on $X$, the $G$-invariant cochains with coefficients in an Abelian group $A$ define a sub-cocomplex $\mathcal{C}^{\bullet}_G$ of the cocomplex $\mathcal{...
5
votes
0answers
109 views

Geometrical meaning of Atiyah-Bredon exact sequence in equivariant cohomology

Let a torus $T=(\mathbb C^*)^n$ act on a topological space $X$, and denote by $X_i$ the union of orbits of dimension $i$ and smaller. Suppose that the equivariant cohomology $H^*_T(X)$ are a free ...
12
votes
1answer
379 views

Calculations of cup products in Bredon cohomology

Let $G$ be a finite group. In [1], Bredon defines an equivariant cohomology theory for $G$-CW complexes $H^*_G(X;M)$. The coefficients are taken in modules over the orbit category of $G$, that is, ...
1
vote
0answers
53 views

Equivariant Formula for High Dimensional Isolated set

The Atiyah-Bott-Berline-Vergne-Witten localization formula says $S^1$ acting on compact manifold $M$ isolated fixing points. And for a closed equivariant form $\omega$, then $$(2\pi)^{-\frac{\dim(M)}...
5
votes
1answer
152 views

Intuition for the construction of the space $M_G=EG\times _G M$

Reference: Atiyah & Bott, The moment map and Equivariant cohomology Question: What could be the motivation and the intuition behind the construction of the space $M_G=EG\times _G M$? When I am ...