Questions tagged [equivariant-cohomology]
The equivariant-cohomology tag has no usage guidance.
177
questions
19
votes
2
answers
797
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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 ...
- 6,722
19
votes
2
answers
3k
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Is there any "deep" relation between the localization theorem of equivariant cohomology and the localization theorem of equivariant K-theory
First let's consider equivariant cohomology: if a compact Lie group $G$ acts on a compact manifold $M$. We have the equivariant cohomology $ H_G(M)$ defined as the cohomology of the cochain complex $((...
- 8,707
18
votes
8
answers
3k
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Reference request: Equivariant Topology
I am teaching a graduate seminar in equivariant topology. The format of the course is that I will give 2-3 weeks of background lectures, then each week a student will present a topic. The students ...
16
votes
2
answers
2k
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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{...
- 843
16
votes
1
answer
790
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"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 $G$-...
- 843
16
votes
1
answer
2k
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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 ...
- 153
16
votes
0
answers
1k
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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 $X$ ...
- 3,519
15
votes
2
answers
1k
views
$RO(G)$-graded homotopy groups vs. Mackey functors
Everything here is model-independent: either take co/fibrant replacements wherever appropriate, or work $\infty$-categorically.
Also, I've looked through other similar MO questions, but I didn't find ...
- 6,039
14
votes
5
answers
2k
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Reading list for Equivariant Cohomology
I was applying equivaraint cohomology, in particular in the symplectic setting, for some time, but I feel like I am missing some nice books/course notes/articles. Could you advise me some literature, ...
- 14k
14
votes
0
answers
516
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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 ...
- 4,304
13
votes
2
answers
1k
views
Microlocalizing Hochschild homology
A recent paper of Ben-Zvi and Nadler gives a general formalism for understanding "dimensions" in sheaf theories. Without getting too far into details, amongst other things, this formalism allows us ...
- 44k
12
votes
1
answer
314
views
Passing from T-equivariant to G-equivariant cohomology
Let G=GLn(ℂ) and let T be a maximal torus. Let X be a topological space with a G-action. My question is: when is the canonical map $$H^*_G(X;\mathbb{Z})\to H^*_T(X;\mathbb{Z})$$ injective?
Some ...
- 8,726
12
votes
1
answer
529
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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, ...
- 35k
11
votes
1
answer
523
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 ...
- 843
11
votes
0
answers
943
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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 ...
- 6,666
10
votes
1
answer
1k
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T-equivariant cohomology of flag variety
Let $X=G/B$ , where $G=GL_n(\mathbb{C}^n)$ and $B$ be the upper triangular matrices. I am curoius about the structure of $H^*_T(G/B)$ which I consider as a $H_T^*(pt)$-module. If we just consider ...
- 849
10
votes
2
answers
709
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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 ...
- 2,273
10
votes
0
answers
177
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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 $O(2n)$, ...
- 9,388
10
votes
0
answers
315
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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 ...
- 332
9
votes
3
answers
722
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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 ...
- 914
9
votes
1
answer
920
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Rational homology and finite group actions
I'm looking for examples of the following phenomena. Let $X$ be a reasonable space (say, a CW complex) and $G$ be a finite group acting on $X$. For all $k \geq 1$, the projection map $X \rightarrow ...
- 93
9
votes
2
answers
860
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Equivariant motivic sheaves
Thanks to the work of Cisinski-Deglise: http://arxiv.org/abs/0912.2110, we now have a triangulated category of `motivic sheaves' available that admits the standard yoga of the six functors.
Is there ...
- 1,595
9
votes
1
answer
664
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Citation: earliest incidence of the Borel localization theorem
The Borel localization theorem in (Borel) equivariant cohomology states that if $T$ is a torus and $M$ a smooth $T$-manifold, with fixed point set $M^T$, then upon localizing the coefficient ring $H^*...
- 2,984
9
votes
1
answer
1k
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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 ...
- 7,549
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 $d$...
- 6,719
9
votes
0
answers
221
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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 $X$. ...
- 343
9
votes
0
answers
387
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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 ...
- 35k
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 ...
- 115k
8
votes
1
answer
505
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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 ...
- 83
8
votes
1
answer
290
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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 $G$-...
- 2,984
8
votes
1
answer
1k
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Equivariant cohomology of finite group actions and invariant cohomology classes
Let $W$ be a finite group acting on a space $X$. In what generality is it true that $H^*_W(X) = H^* (X)^W$? We always have a map $H^*_W(X) \rightarrow H^* (X)^W$, but it is certainly not an ...
- 445
8
votes
2
answers
184
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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 ...
- 293
8
votes
1
answer
927
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-...
- 7,549
8
votes
1
answer
655
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 ...
- 7,549
8
votes
1
answer
469
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
$$NH=\Bbbk[y_i,\partial_{j}]...
- 719
8
votes
1
answer
763
views
Chern-Weil homomorphism and classifying space
Let $G$ be a real or complex Lie group with Lie algebra $\mathfrak g$, and let $\mathbb C[\mathfrak g]$ be the algebra of $\mathbb C$-valued polynomials on $\mathfrak g$. Denote by $$\mathbb C[\...
- 2,719
8
votes
0
answers
697
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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 ...
- 35k
7
votes
1
answer
701
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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 ...
- 21.1k
7
votes
2
answers
448
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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 ...
- 374
7
votes
1
answer
415
views
How does the kernel of the map $\Omega^{\bullet}(X)\rightarrow \Omega^{\bullet}(G\times X)$ relate to equivariant cohomology?
This question may be trivial for experts. Consider a (compact, connected) smooth manifold $X$ and a (compact connected) Lie group $G$ act on $X$. Then we have the action map
$$
\mu: G\times X\...
- 8,707
7
votes
1
answer
622
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Computing the equivariant cohomology of a specific $(\mathbb{Z}/2\mathbb{Z})^2$-space
In the paper On the Castelnuovo-Mumford regularity of the cohomology ring of a group, Symonds describes the following space.
Let $G = (\mathbb{Z}/2\mathbb{Z})^2 = \{1,a,b,ab\}$ be an elementary ...
- 4,850
7
votes
1
answer
350
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Differentials in Weil model for equivariant cohomology
Why should we define the differential in Weil model as follows? I could understand $\sum_{j,k} c_{jk}^i \theta_j \wedge \theta_k$ plays a role in the formula because it is the dual of the structure ...
- 667
7
votes
2
answers
591
views
Equivariant Stratifications of a Variety
Let $X$ be a complex variety acted upon algebraically by a complex torus $T$. Suppose that $\{X_{\beta}\}_{\beta\in S}$ is a finite $T$-equivariant stratification of $X$, so that the $X_{\beta}$ are ...
- 4,870
7
votes
1
answer
250
views
The set of isomorphism classes of Z/nZ-equivariant line bundles over a 2 dimensional Z/nZ-CW complex
Suppose I wish to find the set of isomorphism classes of $\mathbb{Z}/n\mathbb{Z}$-equivariant line bundles over a 2-dimensional, compact $\mathbb{Z}/n\mathbb{Z}$-CW-complex $X$, i.e. $\mathrm{Vect}^{1}...
- 85
7
votes
1
answer
523
views
Naive equivariant transfer
Given a group $G$, a $\mathbb Z$-graded cohomology theory $E^*_G$, and a $n$-sheeted covering $p\colon X \to Y$, I would like a transfer map $$p_!\colon E^*_G X \to E^*_G Y$$ satisfying $$\require{...
- 2,984
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&...
- 44k
7
votes
0
answers
229
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 ...
- 2,719
7
votes
0
answers
176
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
193
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 ...
- 71
7
votes
0
answers
384
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,\...
- 71