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3
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1answer
104 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 ...
5
votes
0answers
121 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
votes
1answer
207 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 ...
4
votes
0answers
107 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,\...
4
votes
1answer
133 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
votes
1answer
161 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
votes
1answer
96 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
votes
0answers
91 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
votes
1answer
448 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
votes
0answers
95 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
117 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
136 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
votes
1answer
226 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
120 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
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0answers
141 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
135 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
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0answers
172 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
votes
2answers
410 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
votes
0answers
149 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
67 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
votes
1answer
381 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
votes
0answers
114 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
175 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
402 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
407 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
86 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
104 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
votes
1answer
157 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
282 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
votes
1answer
369 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
251 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
164 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 $\...
8
votes
0answers
241 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 ...
14
votes
2answers
727 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
88 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 ...
11
votes
1answer
279 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
52 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
131 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 ...
3
votes
0answers
134 views

The localization formula for orbifolds

The ABBV localization formula relates equivariant cohomology of a manifold to the equivariant cohomology of the fixed point set of the action. Wikipedia states the ABBV Localization formula in the ...
6
votes
1answer
349 views

Chern-Weil homomorphism and classifying space

Let $G$ be a real or complex Lie group with Lie algebra $\mathfrak g$, and let $\mathbb C[\mathfrak g]$ be the algebra of $\mathbb C$-valued polynomials on $\mathfrak g$. Denote by $$\mathbb C[\...
10
votes
0answers
286 views

Reference for equivariant Atiyah-Jänich theorem

The equivariant Atiyah-Jänich theorem is an isomorphism $$ [X,F]_G \cong K_G^0(X), $$ where $G$ is a compact Lie group, $X$ is a compact $G$-manifold, $F$ is the space of Fredholm operators on a ...
2
votes
0answers
127 views

The support of a module over a equivariant cohomology ring $H^*_G$ is naturally a subset of $\mathfrak g$

Reference: M. Atiyah, R. Bott, "The moment map and equivariant cohomology," Topology 23 (1984) (Page 5) Let $G= T$ be a torus of dimension $l$. You may assume $G=(\mathbb C^*)^{l}$. Then the ...
6
votes
2answers
431 views

Reference request: an example of Bott residue formula's usage

Could you give me an example of a clear and beautiful application of Bott residue formula in torus-equivariant cohomology (see below)? I found an example calculating a product of Chern classes on ...
1
vote
1answer
239 views

Elementary question: Intuition for equivariant cohomology

A group $G$ acts freely on a manifold $M$, then $H^*_G(M)=H^*(M/G)$. Why is $H^*_G(M)$ a torsion $H^*_G$-module, where $H^*_G=H^*_G(pt)=H^*(BG)$? If $G=T=(S^1)^{n+1}$ is a torus then $H^*_G=H^*...
1
vote
1answer
221 views

Vector bundles and equivariant vector spaces

It seems commonly accepted that most of the results of equivariant geometry for vector spaces yield analog result for vector bundles. In so far as I understand it, the reason for that is the ...
6
votes
1answer
439 views

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 ...
4
votes
0answers
284 views

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 ...
4
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0answers
144 views

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 \...
2
votes
0answers
185 views

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 ...
10
votes
0answers
432 views

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 ...