Questions tagged [equivariant-cohomology]

The tag has no usage guidance.

79 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
16 votes
1 answer
2k views

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 ...
GSM's user avatar
  • 343
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 $X$ ...
Catherine Ray's user avatar
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 ...
Konrad Waldorf's user avatar
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 ...
Sebastian Goette's user avatar
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 $O(2n)$, ...
mme's user avatar
  • 9,263
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 ...
Luuk Stehouwer's user avatar
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 $d$...
domenico fiorenza's user avatar
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 $X$. ...
Just Me's user avatar
  • 343
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 ...
Mark Grant's user avatar
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 ...
Qiaochu Yuan's user avatar
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 ...
Mark Grant's user avatar
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&...
Ben Webster's user avatar
  • 43.9k
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 ...
Hang's user avatar
  • 2,719
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&...
Nicolas Hemelsoet's user avatar
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 ...
kyler's user avatar
  • 71
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,\...
ventania's user avatar
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 ...
Doron Grossman-Naples's user avatar
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 ...
Mark Grant's user avatar
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 ...
Dominic Else's user avatar
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 ...
Saal Hardali's user avatar
  • 7,549
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{...
Matthias Wendt's user avatar
6 votes
0 answers
184 views

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 ...
Peter Crooks's user avatar
  • 4,870
5 votes
0 answers
147 views

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 $G/H\cong S^{2k-1}$ ...
Yun Liu's user avatar
  • 51
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 ...
evgeny's user avatar
  • 1,980
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 \...
Misaka01034's user avatar
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 ...
jdc's user avatar
  • 2,984
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 ...
Haggai Tene's user avatar
4 votes
0 answers
96 views

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 ...
Antoine Labelle's user avatar
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 ...
Motmot's user avatar
  • 293
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 ...
N.B.'s user avatar
  • 757
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 \...
Wille Liu's user avatar
  • 1,056
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 ...
Thomas Rot's user avatar
  • 7,363
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 ...
user3158840's user avatar
4 votes
0 answers
119 views

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 ...
Peter Crooks's user avatar
  • 4,870
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 ...
Vanya Karpov's user avatar
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 ...
EquivariantGuy's user avatar
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 ...
user avatar
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(\...
Alex's user avatar
  • 131
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 ...
Mira's user avatar
  • 129
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 ...
Leonard's user avatar
  • 151
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 ...
aglearner's user avatar
  • 14k
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 $\...
Igor Sikora's user avatar
  • 1,759
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 $\...
B K's user avatar
  • 1,890
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 $G$ ...
Marion's user avatar
  • 577
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^*...
Satoshi  Nawata's user avatar
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 ...
Peter Crooks's user avatar
  • 4,870
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 ...
Peter Crooks's user avatar
  • 4,870
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 ...
sagirot's user avatar
  • 455
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 ...
Bingyu Zhang's user avatar
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 ...
user519810's user avatar