Questions tagged [equivariant-cohomology]
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185 questions
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Torus localisation for F_q-functions?
Torus localisation for equivariant cohomology of manifolds/... is one of the most useful techniques in geometry.
Second, there are strong analogies between sheaf cohomology and the ring of $\mathbf{F}...
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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 ...
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Equivariant Chow groups
I am new to algebraic geometry. I am reading about equivariant Chow groups from the paper "equivariant intersection theory" by Edidin and Graham. I was trying to calculate some basic ...
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Representability of cohomology theories in $\mathbb{A}^1$ homotopy category
I am new to algebraic geometry and $\mathbb{A}^1$ homotopy theory. My question is the following:
I know that the algebraic $K$ theory is reperesenatble in $\mathbb{A}^1$ homotopy category over smooth ...
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Is there a relationship between fusion and S^1-equivariance for spinors on loop space?
A while ago (officially in 1987), Witten conjectured that string structures on a manifold $M$ correspond to "$S^1$-equivariant" (or more precisely $\mathrm{Diff}^+(S^1)$-equivariant) spin ...
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Serre spectral sequence of Borel construction
Let $G$ be finite $p$-group, $X$ be path-connected $G$-space, $E=EG\times_{G}X$ be the Borel construction and $BG$ be the classifying space of $G$. Consider Serre spectral sequence of the fibration
$$...
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Equivariant KR-theory of representation sphere
I would like to say my question first.
Let $G$ be a compact Lie group acting on a good space $X$ in a good way. Let $V$ be a $G$-representation whose real dimension may be less than 8, and let $S^V$ ...
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Equivariant cohomology of fixed points using the localisation theorem
I am trying to understand the Smith-Thom inequality for spaces equipped with an action by a cyclic group and also the case, when it's an equality:
In the following, let $G=\mathbb{Z}/p$, $\mathbb{F}$ ...
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Integral equivariant formality for Hamiltonian T-actions
What is the simplest example of a compact symplectic manifold $M$ with Hamiltonian $T$-action for which the integral $T$-equivariant cohomology is not formal
i.e. $$H_T^*(M,\mathbb{Z}) \not \cong H^*(...
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About Čech cohomology in transformation groups
I'm starting a study about theory of transformation groups and equivariant cohomology, in what I read several times that Čech cohomology is the most compatible with this theory, but until now I haven'...
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Cohomology of equivariant toric vector bundles using Klyachko's filtration
I am trying to understand Klyachko's following description of the cohomology groups of locally free (hopefully more generally of reflexive) sheaves on toric varieties.
Whereas detailed literature ...
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Are the spaces BG for compact connected groups G ind-projective or ind-Kähler?
Let $G$ be a compact connected group, or maybe better its complexification. By thinking about the simplicial Borel space, or using $n$-acyclic $G$-spaces for higher and higher $n$, it's "easy&...
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Recover the $C_k$-action of a cyclic object as from the $S^1$-action on Hochschild chain
$\newcommand{\Fun}{\operatorname{Fun}}$Let $X\in\Fun(\Lambda^\mathrm{op}, \mathcal{C})$ be a cyclic object in the ($\infty$-)symmetric monoidal category $\mathcal{C}$, where $\Lambda$ is the cyclic ...
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Equivariant $K$-theory and proper actions of discrete groups
The work of Lück and Oliver describes the generalization of equivariant $K$-theory to infinite discrete groups. When $X$ is a finite proper $G$-CW complex, there exist Bott isomorphisms $K^n_G(X)\cong ...
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Is anything known about the equivariant homotopy theory of surfaces with the action of a finite subgroup of the mapping class group?
The Nielson realization theorem for a surface says that every finite subgroup of the mapping class group is realized by a finite subgroup of homeomorphisms on the surface. Furthermore, for a genus $g \...
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Equivalent descriptions of equivariant K-theory
I am looking at references for computing $$K_{T}(G/H)$$ where $G$ is a compact connected Lie group with maximal torus $T$, and $H\subset G$ is a corank one Lie subgroup such that $G/H\cong S^{2k-1}$ ...
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What is $H^*(\mathbb{CP}^{2^N-1}/\Sigma_n;\mathbb{Z})$ when $N=\binom{n}{2}$?
$H^*((S^3)^N/\Sigma_n;\mathbb{Q})$ is computed here.
It makes a little more sense to compute $H^*((S^2)^N/\Sigma_n;\mathbb{Q})$ given that global phase is irrelevant. The proof is exactly the same.
...
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Mackey coset decomposition formula
I have a question about following argument I found
in these notes on Mackey functors:
(2.1) LEMMA. (page 6) Let $G$ be a finite group and $J$ any subgroup. Whenever $H$ and $K$ are subgroups of $J$, ...
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Torsion equivariant cohomology of reductive groups
Let $G$ be a reductive group with maximal torus $T$. One knows that the equivariant cohomology ring of a point with rational coefficients is $\mathbb{Q}[X^*(T)]^W$, and also there is an equivariant ...
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Representation-theoretic interpretation of double Schur polynomials
The Schur polynomials
$$s_\lambda(x_1, \ldots, x_n) = \frac{|x_i^{\lambda_j+n-j}|_{1\le i,j\le n}}{|x_i^{n-j}|_{1\le i,j\le n}}$$
naturally appear as polynomial representatives for Schubert classes in ...
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Canonical reference for dictionary between $G$-spaces and fiber bundles over $BG$?
I'm looking for a comprehensive reference (for citation purposes) laying out the basic facts of the equivalence between $G$-spaces and bundles over $BG$ for a discrete group $G$. I'd like it to also ...
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Definition of Cartan Model - Equivariant differential forms
I would like to ask about an equivalence between two definitions for the Cartan Model.
Let $G$ be a connected Lie group, let $\mathfrak{g}$ be its Lie algebra and let $M$ be a $G$-manifold. The Cartan ...
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The “Kunneth-type” morphism in equivariant $K$-theory
Suppose that one has two algebraic varieties with action of a reductive group $G$: say, $X$ and $Y$.
There is an evident Kunneth-type morphism
$K_G(X) \otimes K_G(Y) \to K_G(X \times Y)$,
where the ...
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Equivariant complex $K$-theory of a real representation sphere
Consider the one-point compactification of a $U(n)$-representation $V$, denoted by $S^V$. I want to caclulate $\tilde{K}_\ast^{U(n)}(S^V)$. When $V$ is a complex $U(n)$-representation, we can use the ...
user501794
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Terminology for equivariant homology
The usual $G$-equivariant homology and cohomology groups of a space $X$ with $G$-action are given by the Borel construction:
$$H_\ast^G(X)=H_\ast((X\times EG)/G),$$
$$H^\ast_G(X)=H^\ast((X\times EG)/G)...
- 18.8k
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Computing the equivariant Chern character
Suppose I know the Chern character of an object $F \in D^b(X)$, where $X$ is some smooth complex projective variety with a finite group $G = \mathbb{Z}/m$ acting on it. In $D^b([X/G]) \simeq D^b(X)^G$ ...
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Is the localised $S^1$-equivariant cohomology of the free loop space of a space $X$ isomorphic to that of $X$ itself?
A well-known theorem of Atiyah and Bott states that given a finite dimensional oriented manifold $M$ with circle action, the $S^1$-equivariant cohomology of $M$ (with $\mathbb{Q}$ coefficients) is ...
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Equivariant cohomology of symmetric group acting on a product
Let $X$ be a finite CW-complex. The symmetric group $S_n$ acts on the product $X^{\times n}$ in the obvious way. Let $H^{\bullet}_{S_n}(X^{\times n})$ be the (Borel) equivariant cohomology of this ...
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What is the circle-equivariant cohomology of the real projective plane
Let P denote the real projective plane. It has an action of the circle group S1. (e.g. Let S1 act on the 2-sphere by rotations about an axis, then this action descends to the quotient P). I have a ...
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Equivariant K-theory for products of groups?
Let $X$ be a $(G \times H)$-space. What is known about the connection between the groups $K_G(X)$, $K_H(X)$ and $K_{G \times H}(X)$? The $G$ and $H$ action on $X$ come from the canonical inclusions $G ...
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Polynomiality of the equivariant Euler characteristic of a sheaf tensored with a standard line bundle on the flag variety
Let ${\mathcal B}=Fl(V)$ be the variety of complete flags in an $(m+2n)$-dimensional vector space $V$ over $\mathbb C$. Then we have the standard line bundles $\mathcal{O}(\lambda)$ on $\mathcal B$ ...
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Frobenius pushforward of an equivariant tautological bundle on the flag variety
Let $m,n$ be nonnegative integers. Let $k$ be a field of positive characteristic. Let the vector space $V$ over the field $k$ have basis $e_1,...,e_{m+n}, f_1,..., f_n$. Let $k^*$ act on $V$ by $t \...
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$E^G_\ast(E)$ tensored with the rationals
Lemma 17.19 of Switzer's "Algebraic topology - Homology and Homotopy" states that $E_\ast(F)\otimes\mathbb{Q}$ is isomorphic to $\pi_\ast(E)\otimes\pi_\ast(F)\otimes\mathbb{Q}$. I wanted to ...
user493302
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Equivariant spectra with coefficients
In “The localization of spectra with respect to homology”, Bousfield describes localizations with respect to Moore Spectra. Given a spectrum $E$, and a group $M$, he describes the spectrum with ...
user340793
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Is the equivariant Steenrod algebra useful?
I am a newbie to the field, so please excuse any potential obvious gaps in knowledge. I have been wondering of late about the equivariant (dual) Steenrod algebra in the context of genuine $G = C_p$ ...
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Isomorphism between Weyl and Cartan models as Hom-Tensor Adjunction
Let $M$ be a manifold, $\Omega$ be the de Rham complex of $M$. Let $G$ is a compact Lie group acting on $M$, $\mathfrak g$ its Lie algebra and $W(\mathfrak g) = \Lambda(\mathfrak g^*) \otimes S(\...
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Generalisation of the equivalence between $C^*(H)$ and $C_0(G/H) \rtimes G$; induction of group actions on C*-algebras
There is a well known Morita equivalence between the group C*-algebra $C^*(H)$ and $C_0(G/H) \rtimes G$, where $H$ is a subgroup of $G$. The corresponding equivalence of representations is an ...
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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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Borel vs genuine equivariant cohomology in quantum field theory
A lot of important work in quantum field theory involves Borel equivariant cohomology of certain geometric objects, usually with the goal of computing integrals over some complicated moduli stack. In ...
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Localization for generalized Borel cohomology
For both equivariant de Rham cohomology and equivariant K-theory (in the "naive" or Borel sense), we have localization formulae which allow us to compute this cohomology in terms of the ...
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Comparing cohomology of quotient by algebraic group and Borel subgroup
Let $X$ a variety over an algebraically closed field $k$ (which we can assume to be actually $k=\mathbb{C}$) and $G$ a connected reductive algebraic group acting freely on $X$ (we can actually assume $...
- 1,081
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A connection between equivariant and non-equivariant cohomology of toric variety
Let $X$ be a smooth projective toric variety over $\mathbb{C}$. It is acted by the compact torus $T=(S^1)^n$.
The $T$-equivariant cohomology $H^*_T(X)$ (with coefficients in a field, say) is an ...
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Geometric fixed points of induction spectrum
I was reading the paper "The Balmer spectrum of rational equivariant cohomology theories" of J.P.C. Greenlees and I found the following interesting fact, expressed in Lemma 4.2 and Remark 4....
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If two symplectic toric manifolds are diffeomorphic, are they necessarily equivariantly diffeomorphic?
Suppose $M$ and $N$ are two symplectic toric manifolds. If $M$ is diffeomorphic to $N$, can we deduce that $M$ is equivariantly diffeomorphic to $N$ with respect to their torus actions?
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Endomorphism in the rational stable $O(2)$-equivariant category of the universal space of the family of finite dihedral subgroups
Let $G$ be a compact Lie group. We can define $\mathfrak{F}G$ to be the collection of conjugacy classes of closed subgroups of $G$ whose Weyl group is finite, a bi-invariant metric on $G$ induces a ...
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Deequivariantisation of indecomposable sheaves
Let G be a connected group acting on a space X. All spaces should be reasonable, so e.g. G is a complex affine algebraic group acting on an algebraic variety X, with everything done using the usual ...
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On realizing a topos of sheaves as a topos of equivariant sheaves
This question is motivated by the following example : let $X$ be a variety over a field $k$, with algebraic closure $\bar{k}$. The Galois group $G_k:=\mathrm{Gal}(\bar{k}/k)$ acts on $X_{\bar{k}}:=X\...
- 617
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Cohomology of quotient stack
Let $X$ be an algebraic variety over $\mathbb{C}$ (the ground field is not important but this makes things easier I think) and $G$ an algebraic group acting over it. Let's say we know that there's a ...
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What's the definition of Euclidean density?
In page 19 of the article "Equivariant cohomology with generalized coefficients" the authors say:
Let $V $ be an n-dimensional real vector space, let $v'$ be a non-zero element in $ \wedge^...
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What is the pointed Borel construction of the $0$-sphere?
From what I understand, the Borel construction takes a $G$-space $X$ and produces a topological space $X\times_{G}\mathbf{E}G$―the homotopy quotient $X/\!\!/G$ of $X$ by $G$ in the $\infty$-category ...
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