# 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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### 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 $\mathbb{R}^2$?
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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 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 $S^1$-...
1 vote
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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, $A$...
1 vote
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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 ...
1 vote
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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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### 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, (...
1 vote
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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 ...
1 vote
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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 ...
1 vote