Mathematics

Questions tagged [equivariant-cohomology]

69 questions
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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 should interpret the Borel cohomology $H^*_{G}X$ of a…
Tyrone
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11
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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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$G$-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
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6
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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6
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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
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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
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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
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5
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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 canonical map, the characteristic map , $$…
A. Bellmunt
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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 transformations. Let $\mathbb{Z}_2$ act on $\tilde X\times…
maz
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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,…
Maxime Ramzi
  • 45,386
4
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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
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How Berline–Vergne Localization Formula derives the abstract formula in the fixed point of the action

I am working with the Berline–Vergne localization formula (from Heat Kernels and Dirac Operators): For a compact Lie group $G$ acting on a compact manifold $M$ and an equivariantly closed form $\alpha$ , their theorem (7.13) states that for $X \in…
dani_mrbg
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Equivariant cohomology of three copies of $\mathbb{CP}^1$ joined at the fixed points.

In Goresky, Kottwitz and MacPherson's classic paper , they give the following example of a space whose equivariant cohomology is not free as an $H^*(BS^1)$ -module: Let $K = S^1$ act on $\mathbb{CP}^1 \cong S^2$ by rotation with fixed points at the…
feynhat
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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
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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…
ThePiv
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