Questions tagged [equivariant-cohomology]
66 questions
13
votes
1 answer
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 should interpret the Borel cohomology $H^*_{G}X$ of a…
Tyrone
- 17,539
11
votes
1 answer
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 clarification, but the answers are often from years…
jdc
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6
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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 reasonable thing to consider ?
For instance, a…
Maxime Ramzi
- 45,086
6
votes
2 answers
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 of $G$ equivariant sheaves $\mathcal{Sh}_G (X)$.…
quinque
- 2,722
5
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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 $S^1$-equivariant cohomology is…
Todd
- 161
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5
votes
1 answer
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, $A$ a left $\rho$-module and $K$ a chain complex on…
Daniel Bernoulli
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5
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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 topological space $X$ with an action of $G$ the…
evgeny
- 3,931
5
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1 answer
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 canonical map, the characteristic…
A. Bellmunt
- 1,702
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4
votes
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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 different primes than $2$ (or, more generally, at…
Maxime Ramzi
- 45,086
4
votes
1 answer
$Z_2$ 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 inversion, so $(x,y) \to (-x,-y)$. This action does not…
Physicist
- 167
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3
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Equivariant K-theory of a contractible variety
Let $X$ be an affine algebraic variety (over $\mathbb{C}$). Let $H$ be an algebraic group acting on $X$. Assume also that there exists an action of $\mathbb{C}^\times$ on $X$ commuting with the action of $H$ and contracting $X$ to one point. Let…
Asav
- 293
3
votes
2 answers
Cohomology Isomorphism of Classifying Spaces and Equivalence of Compact Lie Groups
Let $R$ be a commutative ring with unity, and let $G$ and $H$ be compact Lie groups with $BG$ and $BH$ as their respective classifying spaces. If there exists isomorphism $H^j(BH; R) \to H^j(BG; R)$ for all $j\geq 0$, can we conclude that the groups…
ThePiv
- 63
3
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Equivariant Hochschild/Harrison (co)homology
We have the equivariant cohomology theory in topology/geometry, when the topological space has a group
action, equivariant cohomology is a functor that reflects both the topology of the space and the action of the group. For an introduction please…
mathisfun
- 71
3
votes
1 answer
Why does the fixed point locus of a circle action always have even codimension
So I'm studying equivariant cohomology, and I am in particular interested in the case where there is a $U(1)$ action on a Riemannian manifold $\mathcal{M}$. This is generated by some vector field $\xi\in\mathfrak{X}(\mathcal{M})$. The fixed points…
arow257
- 374
3
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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 $\Omega_G(M) := \{ a \in S(\mathfrak{g}^*) \otimes \Omega(M) | a \text{ is invariant} \}$.
First definition of…
Nash-iOS
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