Questions tagged [equivariant-cohomology]

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Action of a group on a manifold: terminology of virtual character in equivariant homology?

I am looking for a proper terminology of the following. Additionally, it can be related to equivariant cohomology and I would like to know if there is any connection. Given an action of a finite group ...
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The Mackey functor $\underline{\pi}_n(X)$

Let $G$ be a finite group and $X$ a pointed $G$-space. The assignment $G/H\to \pi_n(X^H)$ should define a Mackey functor. I am trying to figure out what the transfers and restrictions are. If $H\...
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Find a subgroup $K$ to complete the pullback diagram $G/g_1Hg_1^{-1}\leftarrow G/H\to G/g_2Hg_2^{-1}$.

EDIT: I have realised I made a mistake when decompsoing the morphisms of $\mathscr B_G$. Nevertheless, the question seems to be interesting on its own, so i will leave it. I would also like to cite ...
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Why are quotients by free group actions "well behaved"

Let $M$ be a smooth manifold and $G$ a Lie group, then if $G$ acts smoothly, freely, and properly on $M$ it is a well known result that the quotient $M/G$ is a smooth manifold. In the context of ...
Chris's user avatar
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Quotient of $G$-Homotopy equivalent spaces

In this answer, it is claimed that the naive quotient does not behave well with homotopy equivalences. The example given is $\mathbb{R}\to \text{pt}$ with $G=\mathbb{Z}$ acting by translation on $\...
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Reduced Equivariant Cohomology

I read an article stating that if a finite group $G$ acts on a compact and path connected topological space $M$ and the action has a fixed point $x_0$, then $$ H^p_G(M;\mathbb{Z})\simeq H^p_G(x_0;\...
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Definition of Cartan Model - Equivariant forms

Let $G$ be a connected Lie group and let $\mathfrak{g}$ be its Lie algebra. Let $M$ be a $G$-manifold. The Cartan model of $M$ is the $\Omega_G(M) := \{ a \in S(\mathfrak{g}^*) \otimes \Omega(M) | a \...
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Is this a torus action?

I was constructing an explicit example of a torus action in order to compute the Sullivan model of the associated Borel fibration, and I came up with the following action: $T^2 \times T^2 \to T^2$ ...
groupoid's user avatar
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Equivariant Kunneth Formula

Assume that $\tilde X \to X$ and $\tilde Y\to Y$ are double covers. Then $\mathbb{Z}_2$ acts freely on $\tilde X$ by deck transformations, and likewise it acts freely on $\tilde Y$ by deck ...
maz's user avatar
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Minimal polynomial of $\mathcal{O}(1)\otimes -$-operator

This is somewhat of a follow-up to this question. Take $X= \mathbb{P}^1, G = \operatorname{GL}(2)$. Compute the minimal polynomial of the operator $\mathcal{O}(1)\otimes -$ on $K_G(X)$. How does one ...
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Calculating the equivariant K-group $K_G^*(S^1)$ - where's the mistake?

Let a finite group $G$ act on the circle $S^1$ via a group homomorphism $\varphi \colon G \to S^1$. Let $K = \ker \varphi$. I wish to calculate the equivariant K-theory group $K_G^*(S^1)$. One method ...
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Clarification of G-CW(V)-complex via an example

I am trying to understand the definition of a G-CV(V)-complex given by Costenoble and Warner. It seems to me that there are two different definitions. Let $G$ be a finite group and let V be an ...
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Algorithm for writing an image of a polynomial in a quotient ring in terms of a given basis of the quotient ring.

I need to calculate the equivariant Chern classes of certain vector bundles on the classifying spaces of complex algebraic groups. In order to do this I am looking for a way to do the following ...
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Does an equivariant map (/deformation retract) which induces an isomorphism in cohomology also induce an isomorphism in equivariant cohomology?

Let $G$ be a compact Lie group acting smoothly on two manifolds $M$ and $N$ and suppose we have an equivariant map $f: M \rightarrow N$ which induces an isomorphism in cohomology $f^*: H^*(N) \...
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Equivariant maps up to homotopy

Let $G$ be a topological group and $X$, $Y$ be $G$-spaces. Suppose there is a continuous map $F:X\to Y$ which is $G$-equivariant up to homotopy, namely the two maps $G\times X\to Y$, $(g,x)\mapsto F(...
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Equivariant (co)homology of the rotation action of the circle on the plane

What is the equivariant (co)homology of the rotation action of $S^1=SO(2)$ on $\mathbb{R}^2$?
Yeah's user avatar
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Step in proof on the support of a module in equivariant cohomology

I've been stuck on one step of a proof, and am hoping someone will have a helpful hint/explanation :) Here is the setting:we have two Lie groups $K \hookrightarrow T$ acting on a manifold $M$. We can ...
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A mistake in Grothendieck's Tôhoku paper? Theorem 5.2.1

I was reading the ``Sur quelques points d'alegbre homologique'' English translation when I came across the spectral sequences for equivariant cohomology shown in Theorem 5.2.1 (in the original French ...
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$G$-equivariant quasi-ismorphisms over a field

Let $(A, d_A)$ and $(B, d_B)$ be cochain complexes over a field $\mathbb{k}$, and let $f: A \to B$ be a quasi-isomorphism. It is well-known that over a field, every quasi-isomorphism is chain ...
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Is Tu's new 'Introductory Lectures on Equivariant Cohomology' an effective introduction to equivariant topology?

I went to a number of lectures this summer introducing ideas in equivariant algebraic topology. I was interested in learning more and I found a book, Tu's Introductory Lectures on Equivariant ...
Descartes Before the Horse's user avatar
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Equivariant Cohomology, Homotopy Equivalence and Pushforward

If two spaces $X$ and $Y$ are homotopy equivalent, then their cohomologies are isomorphic $$H(X) \cong H(Y).$$ Is there a similar result for the equivariant cohomology? Given algebraic varieties $X_1, ...
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Proof of $\tilde{K}^*_G((G/H)_+\bigwedge X)\cong \tilde{K}^*_H(X)$

I have been reading the paper "A generalization of the Atiyah-Segal completion theorem" by Adams, Haeberly, Jackowski and May, where I found the Following claim: Let $G$ be finite group, $H\...
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$\mathbb Z_2$-equivariant cohomology of tori

I consider the tori $X=(\mathbb R/ 2 \pi \mathbb Z)^d$ on which the group $\mathbb Z_2$ acts by $x \mapsto -x$ for the nontrivial element of $\mathbb Z_2$. This action has $2^d$ fixed points, which ...
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Equivariant Chern classes and local coefficients

I am trying to understand the basics of equivariant cohomology in view of applications to the field of crystalline topological insulators. At stake in that field is the very explicit situation of the ...
Didier Felbacq's user avatar
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Representation Theory and Equivariant Stable Homotopy Theory

What is a good book/source to understand Representations of a Group in the sense we use it in Equivariant Stable Homotopy Theory? I've read Barry Simon's book on Representation Theory but would like ...
Hili's user avatar
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Forgetful Map from Inclusion of Groups on Equivariant Cohomology

So equivariant cohomology of a topological space $X$ with a $G$-action is given by computing cohomology of the homotopy quotient: $H^{*}_{G}(X) = H^{*}(X \times_G EG)$, where $EG$ is a contractible ...
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A misunderstanding about Sullivan's conjecture

In this old blog post, Akhil Mathew describes the Sullivan conjecture and part of Miller's proof of a special case. There's a point in the beginning which is not clear to me, about $p$-completions at ...
Maxime Ramzi's user avatar
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$G$-spaces vs spaces with a $G$-action

In equivariant homotopy theory, it seems like one tends to consider "genuine" $G$-spaces or $G$-spectra, rather than spaces (spectra) with a $G$-action. My (rather soft) question is : why is that a ...
Maxime Ramzi's user avatar
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Understanding implications of the equivariant Darboux-Weinstein theorem

I am trying to understand the implications of the equivariant Darboux-Weinstein theorem, stated here: The book that states this (Hamiltonian Group Actions and Equivariant Cohomology) gives an example ...
Henry Shackleton's user avatar
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What's the cohomology of the classifying stack $[\text{pt}/G]$?

Let $G$ be an algebraic group. Consider the classifying stack $\left[\text{pt}/G\right]$, where $\text{pt} = \text{Spec } \mathbb{C}$. What's the cohomology of the classifying stack $\left[\text{pt}/...
Yuhang Chen's user avatar
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Singular cohomology of complex projective space

In the article "What is ... Equivariant Cohomology?" by Loring W. Tu (https://arxiv.org/abs/1305.4293), I read that $H^*(\mathbb{C}P^{\infty},\mathbb{R})\simeq \mathbb{R}[u]$, where $H^*()$ denotes ...
IgnoranteX's user avatar
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1 answer
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S1 equivariant forms

I was reading the article ``Equivariant cohomology'' by L.W. Tu. In page 3 (on page 425) he describes any $S^1$ equivariant 2n form on a compact, oriented smooth manifold M is given by $\alpha = \...
Anantadulal paul's user avatar
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1 answer
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What is the $S^1$-equivariant cup product on $S^2$?

Consider the sphere $S^2 = \mathbb{CP}^1$ with the $S^1 = \{ \tau \in \mathbb{C} \mid |\tau| = 1 \}$ action given by $$ \tau \cdot [z_1, z_2] = [\tau ^ k \cdot z_1, z_2] $$ The corresponding $S^1$-...
Todd's user avatar
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What's the definition of weight in localization theorem?

I am currently reading a book on symplectic topology. I may have skipped some pages so find it confusing about the Duistermaat-Heckman theorem. In the book it states that Assume Hamiltonian function $...
FreeLanding45's user avatar
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1 answer
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What are some good references to learn about equivariant homotopy theory?

What are some good references to learn the foundations of equivariant homotopy theory/algebraic topology, for someone who has a background in basic homotopy theory and a tad more advanced algebraic ...
Maxime Ramzi's user avatar
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What is meant by the symbol $\mathbb{R}^2_{\hbar}$?

I am reading some papers in mathematical physics (https://arxiv.org/pdf/1006.0977.pdf) and I came across the following symbol $\mathbb{R}^2_{\hbar}$ I don't recognize nor could I find any background ...
Graphite's user avatar
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Reference Request: Spectral Sequence Relating Bredon and Borel Equivariant Cohomology.

Given a compact Lie group $G$ and a $G$-space $X$, useful invariants may obtained by studying the equivariant cohomology of $X$. There are various equivariant cohomology theories that may be defined, ...
Tyrone's user avatar
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12 votes
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Interpretation of Borel equivariant cohomology.

This question should have a good answer somewhere on here, but as of yet I've been unable to find one. Any links to existing writings would be very welcome. My question relates to how exactly one ...
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Reference request: Representability of multiplicative equivariant cohomology theories

Let $G$ be a topological group, say a compact Lie group, and $e^*_G$ a multiplicative $\mathbb Z$-graded $G$-equivariant cohomology theory defined on $G$–CW complexes. Is there some analogue result to ...
jdc's user avatar
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UCT for equivariant cohomology

I use the definition of Steenrod's equivariant cohomology given in chapter V of the book N. E. Steenrod. Cohomology Operations. No. 50 in Annals of Mathematics Studies. Princeton University Press, ...
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$\pi_{n}(X_{G})$ when $G$ is finite and acts freely on $X$

When $X\in\mathbf{sSet}_{\ast}$ and $G$ a finite group (it may be considered a constant simplicial group) that acts freely on $X$. My question is Can I express $\pi_{n}(X_{G})$ in terms of $\pi_{n}(X)...
Victor TC's user avatar
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Definition of equivariant cohomology

Classically equivariant cohomology is defined as in Wikipedia. I found the following definition in Steenrod's Cohomology Operations in the chapter "Equivariant Cohomology". Here $\rho$ is a group, $A$...
Daniel Bernoulli's user avatar
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On the preservation of group cohomology in certain quotients

Let $G$ be an arbitrary countable group, $X$ a finite, free, proper $G$-CW complex and $C(G)_*$ the induced free, finite $G$-Chain complex. This means that $C(G)_*$ is a finite chain complex, whose ...
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Equivariant cohomology with respect to Lie group and its maximal torus

Let $G$ be a compact connected Lie group, $T \subset G$ its maximal torus and $W=N(T)/T$ its Weyl group. The formula 2.11 of Atiyah, Bott The Moment Map and Equivariant Cohomology states that for any ...
evgeny's user avatar
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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 ...
student's user avatar
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Equivariant flat U(1) bundle on a torus

I am trying to understand equivariant flat $U(1)$ bundles on a torus, $(S^1)^N$. By equivariant, I mean equivariance with respect to the natural action of $(U(1))^N$ on $(S^1)^N$. $G$-equivariant ...
Meer Ashwinkumar's user avatar
2 votes
1 answer
251 views

Clarification about Borel $G$-homology theory

I'm starting reading something about $G$-homology and as one of the first example I encountered the Borel $G$-homology defined as $$ H^G_*(X):= H_*(EG\times_G X)$$ where $H_*$ is any homology theory, (...
Luigi M's user avatar
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1 answer
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Equivariant generalization of $\mathcal{O}(1)$ on $\mathbb{CP}^1$

The connection of the line bundle $\mathcal{O}(1)$ on $\mathbb{CP}^1$ is given by \begin{equation} A=\frac{i}{2}\frac{\overline{z} \, dz-z\,d\overline{z}}{1+|z|^2} \end{equation} This follows since ...
Meer Ashwinkumar's user avatar
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Killing vector field for Witten complex?

I am reading a classical paper by Atiyah Bott "The moment map and equivariant cohomology". In paragraph "Relation with Witten complex" (at the very beginning of this paragraph) they claims that “$W_s$...
quinque's user avatar
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6 votes
2 answers
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Equivariant cohomology via equivariant sheaves

Ordinary cohomology of topological space $X$ are known to be the cohomology of constant sheaf. Question Is there analogous description for equivariant cohomology? More precisely. Consider category ...
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