Questions tagged [equivariant-cohomology]
The equivariant-cohomology tag has no usage guidance.
7
votes
1answer
159 views
Differentials in Weil model for equivariant cohomology
Why should we define the differential in Weil model as follows? I could understand $\sum_{j,k} c_{jk}^i \theta_j \wedge \theta_k$ plays a role in the formula because it is the dual of the structure ...
11
votes
1answer
684 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 ...
5
votes
0answers
73 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 ...
4
votes
0answers
55 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 \...
3
votes
1answer
137 views
Relative equivariant cohomology
Let us assume that $X=\mathbb{R}\times S^1$ is given with a $G=\mathbb{Z}_2$ action that corresponds to the symmetry $(x,e^{i\theta})\mapsto(-x,e^{-i\theta})$. I want to compute the equivariant ...
5
votes
0answers
125 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 ...
11
votes
1answer
224 views
Is there a kind of Poincare duality for Borel equivariant cohomology?
Let $G$ be a finite (or discrete) group, $M$ a $d$-dimensional manifold with smooth $G$-action (I am interested in the case where the action is not free, so $M/G$ is not a manifold). For an Abelian ...
5
votes
0answers
129 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,\...
4
votes
1answer
137 views
Transgression image and Serre spectral sequence for tori
Let $\mathbb{K} \subset \mathbb{T}$ be two tori acting on a topological space $X$ (with all the properties you want). We use the notations $$X_{\mathbb{T}} := (X \times E \mathbb{T}) / \mathbb{T}, \...
2
votes
1answer
165 views
Do Mackey (co)homology functors factor through derived categories? References with details?
Let $G$ be a compact Lie group; I will write $SH_G$ for the (equivariant) stable homotopy category of $G$-spectra (say, with respect to a complete universe; does its choice affect the homotopy ...
0
votes
1answer
99 views
Compute the rational cohomology ring $H^*(M;\mathbb{Q})$ of a product of Lie groups (and their quotients)
As the following product is a bit unfamiliar to me:
How do we compute the rational cohomology ring $H^*(M;\mathbb{Q})$ of the product of Lie groups:
$M=SO(n_1)\times U(n_2)\times SU(n_3)\times (...
2
votes
0answers
93 views
Reference request: explicit equivariant localization formula on toric complete intersections
This post is about an equivariant integration formula in a famous paper https://arxiv.org/pdf/alg-geom/9701016.pdf by Alexander Givental, where the author presented the formula without proof or ...
16
votes
1answer
456 views
“Rotated” version of the Atiyah-Hirzebruch spectral sequence
Let $G$ be a group, $X$ a topological space with $G$-action. For an Abelian group $A$, let $\mathcal{C}^n(X,A)$ be the group of $n$-cochains on $X$ with $A$ coefficients. We can treat this as a $G$-...
2
votes
0answers
97 views
Homotopy orbits and Local Cohomology Theorem
I'm procedding with understanding Greenlees's paper "The Four Approaches to Cohomology Theories with Reality": https://arxiv.org/abs/1705.09365
Problem concerns the section 1.D. Consider the cofiber ...
4
votes
1answer
136 views
Computing the Euler class of a vector bundle
I'm having the following problem: let $T \subset G := SO(2k)$ be the maximal torus acting on $V := \mathbb{R}^{2k}$ by linear transformations on each $2$-dimensional component. Denote by $V_T := (V \...
2
votes
2answers
137 views
Bredon cohomology of a sign representation for a cyclic group of order 4
Yet another question "I compute Bredon cohomology of something and I am not sure, whether it is correct".
So I am taking a sign representation $\sigma$ of cyclic group of order 4, $C_4$. Then I ...
8
votes
1answer
226 views
Equivariant bundles invisible in K-theory and Borel cohomology
For a given topological group $G$ there are natural transformations $$K^* \leftarrow K^*_G \overset a\to H^{**}(EG \times_G -;\mathbb Q)$$ from equivariant K-theory, the first forgetting the $G$-...
3
votes
0answers
123 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 $\...
10
votes
0answers
148 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
0answers
137 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$. ...
6
votes
0answers
176 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 ...
18
votes
2answers
414 views
Allowing $G$-CW complexes to have more general cells
Let $G$ a finite group. I've seen three options discussed for making $G$-cell complexes: in increasing generality, one might allow $X_n$ to be constructed from $X_{n-1}$ by attaching cells of the form ...
2
votes
0answers
157 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$ ...
2
votes
0answers
67 views
On the preservation of equivariant cohomology in certain quotients
Let $G$ be a discrete, countable group and $X$ a finite, free, proper $G$-CW complex, such that the underlying $CW$-structure of $X$ is locally finite. Further, let $C_*(X,G)$ be the induced free-...
7
votes
1answer
398 views
Homotopy theoretic description of homotopy fixed points (and obstructions) for an action of group $G$ on a groupoid $X$
There are several scattered statements about fixed points and obstructions which I'd very much like to see unified in some framework.
To state them let $G$ be a group acting on a connected (1-...
6
votes
0answers
115 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{...
5
votes
1answer
176 views
Equivariant cohomology ring is an integer domain
Let $G$ be a connected compact Lie group and let $V$ be a complex $G$-representation. Denote by $\mathbb{P}(V)$ the projectivization of the vector space $V$. I would like to ask a couple of questions ...
10
votes
2answers
409 views
Relation between affine flag and Grassmannian Steinberg variety
Let $\mathcal{K}=\mathbb{C}((t))$ be the field of formal Laurent series over $\mathbb{C}$, and by $\mathcal{O}=\mathbb{C}[[t]]$ the ring of formal power series over $\mathbb{C}$.
Given a semi-simple ...
7
votes
1answer
411 views
Naive equivariant transfer
Given a group $G$, a $\mathbb Z$-graded cohomology theory $E^*_G$, and a $n$-sheeted covering $p\colon X \to Y$, I would like a transfer map $$p_!\colon E^*_G X \to E^*_G Y$$ satisfying $$\require{...
2
votes
1answer
87 views
Orbit decomposition of the restriction of an equivariant sheaf?
All sets and groups in the question are finite.
In order to understand equivariant sheaves better I'm trying to prove some basic facts from Mackey theory using equivariant sheaves. The main obstacle ...
3
votes
1answer
111 views
Show the Cartan 3-form transgresses to the Killing form in the Weil algebra
Let $G$ be a connected, reductive Lie group, and $W\mathfrak g = (S[\mathfrak g^\vee] \otimes \Lambda[\mathfrak g^\vee],\delta)$ the associated Weil algebra. This is a CDGA equipped with an action of $...
4
votes
1answer
161 views
Functors between categories of equivariant sheaves are equivariant sheaves on the product?
This is a follow up question to this question which remained unanswered (satisfactorily) even after a large bounty. I have made a litlle progress and I have no a more specific question which might be ...
7
votes
1answer
305 views
Computing the equivariant cohomology of a specific $(\mathbb{Z}/2\mathbb{Z})^2$-space
In the paper On the Castelnuovo-Mumford regularity of the cohomology ring of a group, Symonds describes the following space.
Let $G = (\mathbb{Z}/2\mathbb{Z})^2 = \{1,a,b,ab\}$ be an elementary ...
8
votes
1answer
372 views
Interactions (functors) between equivariant sheaves for different groups?
Let $G$ be a finite group and $k$ a field (alg. closed char 0 for simplicity).
To every $G$ set $X$ we can assign the category of $G$-equivariant sheaves of $k$-vector spaces $Sh_G(X)$. It is ...
2
votes
2answers
252 views
How is the equivariant cohomology of a space related to the cohomology of the corresponding associated bundle
Let $X$ be a manifold with a left $G$-action, and let $\Sigma$ be a Riemann surface. How is the equivariant cohomology $H^*_G(X)$ of $X$ related to the de Rham cohomology of the associated bundle $H^*(...
3
votes
1answer
170 views
Equivariant characteristic classes on $\mathbb{P}^n$
Let $T=(\mathbb{C}^*)^n$ act on $\mathbb{P}^n$ torically by
$$t.[x_0:\dots:x_n]=[x_0\;:\;t_1x_1\;:\;\ldots \;:\;t_nx_n]$$
I would like to know an expression for
the equivariant Chern character $\...
8
votes
0answers
249 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 ...
14
votes
2answers
756 views
Equivariant cohomology vs. invariant cohomology vs. cohomology of quotient space
Given a space $X$ and an action of a group $G$ on $X$, the $G$-invariant cochains with coefficients in an Abelian group $A$ define a sub-cocomplex $\mathcal{C}^{\bullet}_G$ of the cocomplex $\mathcal{...
5
votes
0answers
90 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 ...
11
votes
1answer
290 views
Calculations of cup products in Bredon cohomology
Let $G$ be a finite group. In [1], Bredon defines an equivariant cohomology theory for $G$-CW complexes $H^*_G(X;M)$. The coefficients are taken in modules over the orbit category of $G$, that is, ...
1
vote
0answers
53 views
Equivariant Formula for High Dimensional Isolated set
The Atiyah-Bott-Berline-Vergne-Witten localization formula says
$S^1$ acting on compact manifold $M$ isolated fixing points. And for a closed equivariant form $\omega$, then
$$(2\pi)^{-\frac{\dim(M)}...
5
votes
1answer
133 views
Intuition for the construction of the space $M_G=EG\times _G M$
Reference: Atiyah & Bott, The moment map and Equivariant cohomology
Question: What could be the motivation and the intuition behind the construction of the space $M_G=EG\times _G M$? When I am ...
3
votes
0answers
140 views
The localization formula for orbifolds
The ABBV localization formula relates equivariant cohomology of a manifold to the equivariant cohomology of the fixed point set of the action. Wikipedia states the ABBV Localization formula in the ...
6
votes
1answer
360 views
Chern-Weil homomorphism and classifying space
Let $G$ be a real or complex Lie group with Lie algebra $\mathfrak g$, and let $\mathbb C[\mathfrak g]$ be the algebra of $\mathbb C$-valued polynomials on $\mathfrak g$. Denote by $$\mathbb C[\...
11
votes
0answers
295 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 ...
2
votes
0answers
127 views
The support of a module over a equivariant cohomology ring $H^*_G$ is naturally a subset of $\mathfrak g$
Reference: M. Atiyah, R. Bott, "The moment map and equivariant cohomology," Topology 23 (1984) (Page 5)
Let $G= T$ be a torus of dimension $l$. You may assume $G=(\mathbb C^*)^{l}$. Then the ...
6
votes
2answers
435 views
Reference request: an example of Bott residue formula's usage
Could you give me an example of a clear and beautiful application of Bott residue formula in torus-equivariant cohomology (see below)?
I found an example calculating a product of Chern classes on ...
1
vote
1answer
239 views
Elementary question: Intuition for equivariant cohomology
A group $G$ acts freely on a manifold $M$, then $H^*_G(M)=H^*(M/G)$.
Why is $H^*_G(M)$ a torsion $H^*_G$-module, where $H^*_G=H^*_G(pt)=H^*(BG)$?
If $G=T=(S^1)^{n+1}$ is a torus then $H^*_G=H^*...
1
vote
1answer
222 views
Vector bundles and equivariant vector spaces
It seems commonly accepted that most of the results of equivariant geometry for vector spaces yield analog result for vector bundles.
In so far as I understand it, the reason for that is the ...
6
votes
1answer
443 views
Geometric construcion of Proj as a quotient by a $\mathbb{G}_m$ action
I'm trying to translate the Proj construction as a kind of quotient by a $\mathbb{G}_m$ action. Here's what I have so far:
Let $X=Spec A$ be an affine scheme (after this case is setteled I imagine it ...
