Questions tagged [equivariant-cohomology]

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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 ...
Nash-iOS's user avatar
3 votes
0 answers
80 views

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 ...
Vanya Karpov's user avatar
4 votes
1 answer
169 views

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 ...
Shakuntala's user avatar
2 votes
0 answers
97 views

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)...
John Pardon's user avatar
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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 $D^b([X/G]) \simeq D^b(X)^G$ ...
alg_et_geom's user avatar
5 votes
2 answers
342 views

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 ...
Eugenio Landi's user avatar
3 votes
0 answers
131 views

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 ...
EquivariantGuy's user avatar
5 votes
1 answer
310 views

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 ...
Peter McNamara's user avatar
4 votes
1 answer
206 views

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 ...
Motmot's user avatar
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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 $\mathcal B$ ...
IntegrableSystemsEnthusiast's user avatar
4 votes
1 answer
267 views

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 \...
IntegrableSystemsEnthusiast's user avatar
4 votes
1 answer
151 views

$E^G_\ast(E)$ 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 ...
user avatar
3 votes
0 answers
112 views

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 ...
user avatar
2 votes
0 answers
152 views

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 $G = C_p$ ...
abelian_cat's user avatar
3 votes
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114 views

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(\...
Alex's user avatar
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8 votes
2 answers
164 views

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 ...
Motmot's user avatar
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0 answers
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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 ...
Light man's user avatar
6 votes
0 answers
163 views

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 ...
Doron Grossman-Naples's user avatar
2 votes
0 answers
104 views

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 ...
Doron Grossman-Naples's user avatar
1 vote
1 answer
156 views

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 $...
Tommaso Scognamiglio's user avatar
1 vote
1 answer
201 views

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 ...
asv's user avatar
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2 votes
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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....
N.B.'s user avatar
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1 answer
173 views

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?
Li Yu's user avatar
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4 votes
0 answers
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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 ...
N.B.'s user avatar
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5 votes
3 answers
632 views

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 ...
Peter McNamara's user avatar
5 votes
1 answer
233 views

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\...
Adrien MORIN's user avatar
1 vote
1 answer
631 views

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 ...
Tommaso Scognamiglio's user avatar
0 votes
0 answers
178 views

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^...
Samia's user avatar
  • 129
3 votes
1 answer
252 views

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 ...
Emily's user avatar
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3 votes
0 answers
150 views

Justification for the definition of equivariant curvature

Let $G$ be a compact Lie group which act on a smooth manifold $M$. Let $\mathbb{C}[\mathfrak{g}] \otimes \mathcal{A}$ be the algebra of polynomial maps from $\mathfrak{g}$ to $\mathcal{A}(M),$ we ...
Samia's user avatar
  • 129
10 votes
0 answers
168 views

A minimal $\mathbb Z/2$-invariant Morse function on $U(n)$

Consider the group $U(2n)$ of unitary matrices. This has two standard and important homomorphic involutions. The most famous one $A \mapsto \overline A$ is complex conjugation, with fixed-set $O(2n)$, ...
mme's user avatar
  • 8,910
2 votes
0 answers
209 views

Equivariant line bundles and connections

Equivariant line bundle isomorphism classes are classified by the equivariant cohomology group $H^2_{P}(X;\mathbb{Z})$ and let us take $P$ to be finite abelian and $X$ a finite dimensional CW-complex ...
Time suspect's user avatar
7 votes
1 answer
532 views

Cohomology of quotient by free action

Let $G$ be a finite group. Let $G$ act freely on a CW-complex $X$. I heard that the following fact is true. Claim. The canonical map $H^*(X/G,F)\to H^*(X,F)^G$ is an isomorphism, where $F$ is a field ...
asv's user avatar
  • 20.3k
3 votes
1 answer
235 views

Upgrading various algebro-geometric cohomology theories to be equivariant

I'm wondering what the "right" notion of equivariant cohomology is for something like étale cohomology or coherent cohomology, stuff which is expressible as derived functors of global ...
xir's user avatar
  • 1,712
8 votes
1 answer
470 views

Borel equivariant homology of a suspension

Let $G$ be a discrete group. For a $G$-CW complex $X$, let $H^G_{\bullet}(X)$ denote the Borel equivariant homology of $X$. There are also relative versions of this. Here's my question. Let $X$ be ...
Sarah's user avatar
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6 votes
0 answers
180 views

Borel equivariant cohomology operations

Fix a group $G$. For an abelian coefficient group $A$ let $H^*_G(-;A):=H^*(EG\times_G-;A)$ be the Borel cohomology with $A$ coefficents. This is a functor from $G$-spaces to graded abelian groups ...
Mark Grant's user avatar
1 vote
1 answer
213 views

divisors in non-compact toric varieties

Let $X$ be a smooth quasi-projective toric variety of dimension $n$ over $\mathbb C$. Take it to be non-compact, so its fan is not complete. (A good example to keep in mind is a toric Calabi-Yau.) If ...
jj_p's user avatar
  • 481
7 votes
0 answers
181 views

Does the Hodge decomposition hold for equivariant differential forms?

Let $M$ be a Riemannian manifold. The Hodge decomposition tells that $$ \Omega^*(M) = \mathrm{im} \ d \oplus \mathrm{im} \ d^* \oplus \mathscr H^*(M) $$ where $d^*$ is the adjoint operator of the ...
Hang's user avatar
  • 2,661
6 votes
1 answer
343 views

What is equivariant chains on a representation sphere?

For a finite group $G$ and a finite-dimensional real representation $V$ of $G$, denote by $S^V$ the one-point compactification of $V$, with basepoint at infinity. What is the reduced chain complex $...
John Pardon's user avatar
  • 17.9k
9 votes
0 answers
119 views

An overcomplex of the Cartan model in equivariant cohomology?

Given any graded vector space $V^\bullet$ and any degree 1 linear operator $d\colon V^\bullet\to V^{\bullet+1}$, one gets a complex $(\ker(d^2),d)$. Moreover, if $V^\bullet$ is a graded algebra and $d$...
domenico fiorenza's user avatar
3 votes
0 answers
61 views

Relation of $KR(X)$ and $K(Y)$ for $X\to Y$ a $C_2$ principal bundle

It is an important property of usual equivariant $K$-theory that $K_G(X)\cong K(X/G)$ whenever $G$ acts freely on $X$. What can be said about $KR(X)$ when the action of $C_2$ on $X$ is free? In the ...
Leonard's user avatar
  • 161
7 votes
1 answer
244 views

The set of isomorphism classes of Z/nZ-equivariant line bundles over a 2 dimensional Z/nZ-CW complex

Suppose I wish to find the set of isomorphism classes of $\mathbb{Z}/n\mathbb{Z}$-equivariant line bundles over a 2-dimensional, compact $\mathbb{Z}/n\mathbb{Z}$-CW-complex $X$, i.e. $\mathrm{Vect}^{1}...
Time suspect's user avatar
6 votes
1 answer
213 views

An induction formula for spectral Mackey functors, and a fake proof

I'm trying to get a grasp of Barwick's model for genuine $G$-spectra, that is, spectral Mackey functors 1. There's a classical formula about induction, that should be easy to prove, that I was trying ...
Maxime Ramzi's user avatar
  • 11.3k
7 votes
0 answers
165 views

Intersection numbers via residue formula

$\newcommand{\sslash}{\mathbin{/\mkern-7mu/}}$With a friend we are trying to understand residue formulas in the article "Cohomology pairings on singular quotients in geometric invariant theory&...
Nicolas Hemelsoet's user avatar
8 votes
1 answer
393 views

Equivariant (co)homology of flag manifolds, convolution algebra and nil hecke algebra?

For a complex reductive group $G$ and its Borel subgroup $B$, it seems to be well-known that the equivariant homology group $H^G_*(G/B\times G/B)$ forms a nil-Heck algebra $$NH=\Bbbk[y_i,\partial_{j}]...
Cubic Bear's user avatar
4 votes
1 answer
186 views

Bredon cohomology of a permutation action on $S^3$

I've seen a couple of similar questions asking to verify computations of Bredon cohomology here and here, so I will ask one such question myself. Let $\mathbb{Z}/2$ act on $S^3\subset \mathbb{C}^2$ by ...
Grisha Taroyan's user avatar
5 votes
1 answer
272 views

Equivariant cohomology of a semisimple Lie algebra

Suppose $\mathfrak{g}$ is a real Lie algebra integrating to the connected Lie group $G$. One may consider the $G$-equivariant cohomology of $\mathfrak{g}$ ($\mathfrak{g}^*$) where the $G$-action is ...
user18063's user avatar
  • 461
1 vote
0 answers
353 views

Cellular chain complex of $G$-CW-complexes & their differentials

I not completly understand EXAMPLE 2.31 (page 19) dealing with homology of $G$-CW-complexes. Source: http://www.ltcc.ac.uk/media/london-taught-course-centre/documents/LTCC-notes-Lecture3-2019.pdf ...
user267839's user avatar
  • 4,956
8 votes
3 answers
529 views

Homotopy group action and equivariant cohomology theories

Many of the introductory notes on generalized equivariant cohomology theories assume that one is working over the category of $G$-spaces or $G$-spectra. However, one thing that concerns me is that the ...
Arkadij's user avatar
  • 904
2 votes
0 answers
69 views

Analog of Cartan model for equivariant homology

Let $X$ be a manifold, acted on by a Lie group $G$. (For example $X$ real-even-dimensional acted on by $G=U(1)$ with only finitely many isolated fixed points.) The Cartan model for $G$-equivariant ...
jj_p's user avatar
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