Questions tagged [equivariant-cohomology]

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equivariant cohomology with discrete group action

As far as I know, the equivariant cohomology can be regarded as the generalisation of de Rham cohomology with group action on manifolds. From the literature, the group action is Lie group type. I am ...
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What does it mean that a $G$-space embeds into its homotopy quotient as the fiber over the basepoint of the classifying space of $G$?

Let $G$ be a topological group and $M$ be a $G$-space. Let $EG \rightarrow BG$ be a universal $G$-bundle and let $M_G$ be the homotopy quotient $(EG \times M)/G$. What does it mean that $M$ embeds in $...
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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 $\mathbb{R}^2$?
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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}/...
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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 ...
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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 = \...
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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 $S^1$-...
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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 $...
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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 ...
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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 ...
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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, ...
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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 ...
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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)...
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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, $A$...
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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 ...
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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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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 ...
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2 votes
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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, (...
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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 ...
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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 “$W_s$...
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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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Equivariant cohomology, identity

I'm studying equivariant cohomology on three references: Szabo's review about equivariant localization (S); Libine's note on equivariant cohomology (L); Berline, Getzler, Vigne's book "Heat Kernels ...
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Why the equivariant volume of a non-compact space can be finite?

I am very confused with equivariance (equivariant cohomology etc). In specific when one tries to evaluate the equivariant volume of, say, $\mathbb{R}^2$ (with coordinates $x,y$) one finds that it is $...
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Equivariant Cohomology of homotopy equivalent spaces

Let $V$ is contractible space with $T$ torus action, then can I say their equivariant cohomology (in Borel sense) are equal ? i.e for $\bullet = point$ , $H_T^*(V)=H_T^*(\bullet)$ ?
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$Z_2$ Equivariant K-theory of $S^1$

I am interested in the $\mathbf{Z}_2$ equivariant K-theory of $S^1$, but I cannot find any good references or methods to calculate it with the action I have in mind. The action on $S^1$ is an ...
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ordinary cohomology from equvariant cohomology

Is it possible that the ordinary cohomology of a space can be obtained from its equivariant cohomology? action is algebraic torus action and space is nonsingular complete complex algebraic variety ...
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Proving one version of equivariant formality

Let $G$ be a compact, connected Lie group acting smoothly on a compact, connected and oriented smooth manifold $M$. We denote by $H_G^*(M)$ the corresponding equivariant cohomology. We have a ...
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Explanation for a line from a MathOverflow answer

Sometimes I see questions answered on MathOverflow in such a way that I don't really understand the answers. Sometimes I work out what they mean, and other times I can't. I'd like to ask for more ...
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