Questions tagged [equivariant-cohomology]
The equivariant-cohomology tag has no usage guidance.
79
questions with no upvoted or accepted answers
16
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answer
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The homology of the orbit space
Suppose we have an acyclic group $G$ and let $X$ be a contractible CW-complex such that $G$ acts freely on $X$ (we do not suppose that the action is proper).
Is there a way to understand the homology ...
16
votes
0
answers
1k
views
How is an $S^1$-equivariant elliptic cohomology theory affected as we continuously vary the underlying elliptic curve?
Grojnowski constructs a $S^1$-equivariant cohomology theory over a complex elliptic curve $E$, designed to trivially satisfy: $$E^*_{S^1}(pt) = E$$
The functor $E^*_{S^1}(-)$ takes in a space $X$ ...
14
votes
0
answers
516
views
Reference for equivariant Atiyah-Jänich theorem
The equivariant Atiyah-Jänich theorem is an isomorphism
$$
[X,F]_G \cong K_G^0(X),
$$
where $G$ is a compact Lie group, $X$ is a compact $G$-manifold, $F$ is the space of Fredholm operators on a ...
11
votes
0
answers
932
views
Equivariant and orbifold Chern classes
Edit. After thinking about this problem a bit longer, I am not so sure anymore that the Bredon cohomology proposed by Adem and Ruan gives me the invariants I am looking for. I have therefore moved ...
10
votes
0
answers
177
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 $O(2n)$, ...
10
votes
0
answers
313
views
Is there a spectral sequence of Atiyah's topological KR-theory that can be used to compute basic examples?
For Segal's complex $G$-equivariant $K$-theory, it is well-known that there is an Atiyah-Hirzebruch spectral sequence. If say $G$ a finite group and $X$ a finite CW-complex, the second page of this ...
9
votes
0
answers
122
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 $d$...
9
votes
0
answers
220
views
Fixed-points of a topological circle action
Suppose the circle group $G = S^1$ acts on $X$.
If $X$ is a closed smooth manifold (and the action is smooth), then we know the fixed-points $X^G$ are a disjoint union of smooth submanifolds of $X$. ...
9
votes
0
answers
387
views
Equivariant obstruction theory done wrong
Let $G$ be a finite group, and let $p:E\to B$ be a $G$-fibration with fibre $F$. The correct framework for studying equivariant sections of $p$ is Bredon cohomology. With the right definitions, most ...
9
votes
0
answers
534
views
Twisted equivariant modular forms?
I'd like to know where I can find information about a class of objects which I think deserve to be called twisted equivariant modular forms. Let me guess a definition, indicate how it can be made more ...
8
votes
0
answers
695
views
Finite generation of equivariant cohomology rings
Let $G$ be a finite group acting on a topological space $X$. (In my applications, $X$ is the classifying space of a compact Lie group.) Let $H^\ast(-)=H^\ast(-;\mathbb{Z}/2\mathbb{Z})$ denote mod 2 ...
7
votes
0
answers
181
views
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&...
7
votes
0
answers
220
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 ...
7
votes
0
answers
175
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&...
7
votes
0
answers
191
views
Equivariant L-infinity structure associated to a DGBV algebra
Let $V$ be a differential graded Batalin-Vilkovisky (DGBV) algebra over say $\mathbb{C}$. This means that $V$ is a commutative differential graded algebra (CDGA), with differential $\partial$ of ...
7
votes
0
answers
380
views
Equivariant De Rham theorem for orbifolds
Recall that the "classical" equivariant De Rham theorem states that, for a compact Lie group $G$ acting on a compact smooth manifold $M$,
$$H_G^*(M,\mathbb{R})\cong H^*(\Omega_G(M)),$$
where $H_G^*(M,\...
6
votes
0
answers
214
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 ...
6
votes
0
answers
208
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 ...
6
votes
0
answers
225
views
A Poincare dual spectral sequence for equivariant cohomology: extension to generalized cohomology?
This question is a follow-up to my previous question:
"Rotated" version of the Atiyah-Hirzebruch spectral sequence
In that question, I discussed two different spectral sequences for ...
6
votes
0
answers
300
views
Geometric interpretation of a formula for the induced character (fix point localization?)
Let $H < G$ be a subgroup of a finite group $G$. Let $X:=G/H$ and $\mathcal{F} \in Sh_G(X)$ be an equivariant sheaf on $X$ (w.r.t. left multiplication) associated to a finite dimensional ...
6
votes
0
answers
141
views
Binary forms and equivariant derived category
One of the classical questions in invariant theory is the classification of binary forms, i.e., the description of polynomial invariants of the ${\rm SL}_2(\mathbb{C})$-action on ${\rm Sym}^d \mathbb{...
6
votes
0
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184
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A Property of Generalized Equivariant Cohomology
Let $G_i$ be a compact Lie group, $i=1,2$, and let $E_{G_i}^*$ be a $\mathbb{Z}$-graded complex-oriented $G_i$-equivariant generalized cohomology theory with commutative products. Let $X_i$ be a ...
5
votes
0
answers
147
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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 $G/H\cong S^{2k-1}$ ...
5
votes
0
answers
143
views
Geometrical meaning of Atiyah-Bredon exact sequence in equivariant cohomology
Let a torus $T=(\mathbb C^*)^n$ act on a topological space $X$, and denote by $X_i$ the union of orbits of dimension $i$ and smaller. Suppose that the equivariant cohomology $H^*_T(X)$ are a free ...
5
votes
0
answers
308
views
Relation of BRST model of equivariant cohomology and BRST cohomology?
I'm now reading Kalkman's paper "BRST model for equivariant cohomology and representation for equivariant Thom class". And I've seen his definition for BRST model is
$B=W(\mathfrak{g})\otimes \...
5
votes
0
answers
190
views
Example request: equivariant formality versus formality for homogeneous spaces
Recall that a continuous action of a compact Lie group $T$ on a space $X$ is said to be equivariantly formal if the Borel equivariant cohomology $H^*(X \times_T ET;\mathbb Q)$ surjects through ...
5
votes
0
answers
303
views
extension of cohomology theories
In Rudyak's book it is proved (3.22 page 160) that every additive cohomology theory (in the sense of Eilenberg-Steenrod) defined on the category of finite dimensional pointed CW complexes can be ...
4
votes
0
answers
96
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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 ...
4
votes
1
answer
323
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 ...
4
votes
0
answers
61
views
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 ...
4
votes
0
answers
82
views
Formality of fixed points in the equivariant localisation
Let $X$ be a complex algebraic variety equipped with an algebraic $\mathbf{C}^{\times}$-action.
The Borel construction gives a map $f: \mathrm{E}\mathbf{C}^{\times}\times^{\mathbf{C}^{\times}}X\to \...
4
votes
0
answers
280
views
The localization formula for orbifolds
The ABBV localization formula relates equivariant cohomology of a manifold to the equivariant cohomology of the fixed point set of the action. Wikipedia states the ABBV Localization formula in the ...
4
votes
0
answers
418
views
Localization in equivariant cohomology theory for groups other than ($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 ...
4
votes
0
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119
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Computing the Product Structure in Equivariant Cohomology via a Stratification and Thom-Gysin
Let $X$ be a smooth complex projective variety acted upon algebraically by a complex torus $T$. Suppose that $\{X_{\beta}\}_{\beta\in B}$ is a finite $T$-invariant stratification of $X$ into smooth ...
3
votes
0
answers
102
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 ...
3
votes
0
answers
161
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 ...
3
votes
0
answers
124
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 ...
3
votes
0
answers
125
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(\...
3
votes
0
answers
159
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 ...
3
votes
0
answers
63
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 ...
3
votes
0
answers
226
views
What is rigidity of Hirzebruch, and Witten genera?
I would like to find some good references (or any insight) that would help me to understand a few articles mentioning rigidity of Hirzeburch genus. One of the consequences of this phenomenon is that ...
3
votes
0
answers
166
views
Computations of Bredon homology of $S(1+\sigma)$ with Universal Coefficient S.S
What I am trying to do is to compute $\mathbb{Z}$-graded Bredon homology of $S(1+\sigma)$ over $Q\times\Sigma_2$, where
$Q$ is a cyclic group of order 2
$\sigma$ is its real sign representation
$\...
3
votes
0
answers
169
views
Finding generators of equivariant cohomology
Let $(M,\omega)$ be a symplectic manifold with symplectic form $\omega$, carrying a Hamiltonian action of a compact connected Lie group $G$ with moment map $\mu:M\to \mathfrak{g}^\ast$, where $\...
3
votes
0
answers
318
views
Localization of the pushforward in equivariant cohomology
I am reading Nekrasov's paper and in page 2 he considers the $G \times T^2$ equivariant cohomology of the (compactified) moduli space $\tilde{M_k}$ of $U(N)$ instantons on $\mathbb{C}^2$. Here $G$ ...
3
votes
0
answers
353
views
J-function of cotangent bundle of complete flag variety
Givental and Kim showed that the $J$-function of the complete flag variety $Fl_n=SL_{n}/B$ becomes an eigenfunction of the Toda Hamiltonian. How about the $J$-function of the cotangent bundle $T^*...
3
votes
0
answers
148
views
Equivariant Poincare Series of Based Loop Group of SU(2)
Let $\Omega SU(2)$ denote the based loop group of $SU(2)$, and consider the action of $S^1$ on $\Omega SU(2)$ as a maximal torus of $SU(2)$. (This is not the "loop rotation" action.) Is there an ...
3
votes
0
answers
216
views
Equivariant Cohomology of Non-Compact Spaces via Fixed Points
Let $T$ be a complex torus and $X$ a smooth quasi-projective $T$-variety with finitely many fixed points. Denote by $\varphi:H_{T}(X)\rightarrow H_{T}(X^T)$ the map on equivariant cohomology induced ...
2
votes
0
answers
125
views
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 ...
2
votes
0
answers
68
views
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 ...
2
votes
0
answers
63
views
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 ...