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Questions tagged [equivariant-cohomology]

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Weil Model of Equivariant Cohomology

I was reading this paper, and was stuck on a supposedly trivial calculation at page 13. I have trouble understanding the authors' calculation of $d_X$. The authors claimed $D\lambda a= D(a-i(X)a\...
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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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Reference for the Cartan Model for Equivariant Cohomology

I would like references for the Cartan Model for Equivariant Cohomology which does not use supergeometry (like in the book by Berline, Getzler and Vergne). Thanks in advance.
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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}(...
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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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Equivariant Cohomology and Mayer Vietoris sequence [closed]

I'm reading this article upon topological field theory and I'm a bit confused about the way he compute equivariant cohomology of $S^2$ wrt $\mathrm{U}(1)$, i.e. $H^\bullet_{S^1}(S^2)$. You can find ...
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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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Computing $\text{Ext}$ between two coherent sheaves in equvariant category

This is a basic question about equivariant category of coherent sheaves, but I can't sort it out. Suppose we have an algebraic variety $X$ and consider $Y=X\times X$. Then we have an obvious action ...
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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 ...