Questions tagged [equivariant-cohomology]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
1 vote
1 answer
124 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 $...
user avatar
0 votes
0 answers
38 views

String partition function

Hey I am a bit confused because when reading physics papers I encounter the free energy as a generating function of topological invariants as integrals over certain compactified moduli spaces. For ...
user avatar
  • 1
1 vote
1 answer
161 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 ...
user avatar
  • 19k
2 votes
0 answers
82 views

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....
user avatar
  • 645
2 votes
1 answer
151 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?
user avatar
  • 113
4 votes
0 answers
57 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 ...
user avatar
  • 645
4 votes
3 answers
591 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 ...
user avatar
5 votes
1 answer
195 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\...
user avatar
1 vote
1 answer
562 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 ...
user avatar
0 votes
0 answers
102 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^...
user avatar
  • 425
3 votes
1 answer
188 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 ...
user avatar
  • 6,346
3 votes
0 answers
129 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 ...
user avatar
  • 425
10 votes
0 answers
138 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 $O(2n)$, ...
user avatar
  • 7,891
0 votes
0 answers
105 views

Theorem 7.13 (localization theorem) in the book: Heat kernels and Dirac operator

I'm trying to understand theorem 7.13 in the book Heat kernels and Dirac operators which says the following: Theorem 7.13: Let $G$ be a compact Lie group acting on a compact manifold $M$. Let $\alpha$...
user avatar
  • 425
2 votes
0 answers
199 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 ...
user avatar
6 votes
1 answer
413 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 ...
user avatar
  • 19k
3 votes
1 answer
225 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 ...
user avatar
  • 1,369
8 votes
1 answer
423 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 ...
user avatar
  • 83
5 votes
0 answers
161 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 ...
user avatar
  • 33.2k
1 vote
1 answer
169 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 ...
user avatar
  • 471
7 votes
0 answers
138 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 ...
user avatar
  • 2,611
6 votes
1 answer
318 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 $...
user avatar
  • 17.5k
9 votes
0 answers
105 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 $d$...
user avatar
3 votes
0 answers
54 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 ...
user avatar
  • 61
7 votes
1 answer
219 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}...
user avatar
5 votes
1 answer
187 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 ...
user avatar
  • 8,025
7 votes
0 answers
144 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&...
user avatar
8 votes
1 answer
314 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}]...
user avatar
4 votes
1 answer
165 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 ...
user avatar
5 votes
1 answer
211 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 ...
user avatar
  • 411
1 vote
0 answers
227 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 ...
user avatar
8 votes
3 answers
454 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 ...
user avatar
  • 864
2 votes
0 answers
61 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 ...
user avatar
  • 471
3 votes
0 answers
210 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 ...
user avatar
  • 13.2k
3 votes
0 answers
143 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 $\...
user avatar
  • 1,315
2 votes
1 answer
152 views

Equivariant Coefficient ring action on singular cohomology

Let $X$ be a manifold acted on by a Lie group $G$. The $G$-equivariant cohomology of $X$ with coefficients in a ring $\mathcal{R}$ is defined as the cohomology ring $$ H_G^*(X; \mathcal{R}) := H^*(X_G;...
user avatar
  • 1,157
13 votes
5 answers
1k views

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, ...
user avatar
  • 13.2k
5 votes
1 answer
287 views

Equivariant cohomology algebra of toric variety

Let $X$ be a complex projective and smooth toric variety of complex dimension $n$. It is acted by the real torus $T=(S^1)^n$. Is it true that the $T$-equivariant cohomology $H^*_T(X,\mathbb{Z})$ ...
user avatar
  • 19k
4 votes
1 answer
228 views

Transition equations for double Schubert polynomials

For $w$ a permutation, the associated (ordinary) Schubert polynomial $\mathfrak{S}_w(\textbf{x})$ is a multivariate polynomial that represents for the cohomology class of the Schubert variety $X_w$ in ...
user avatar
  • 1,721
2 votes
1 answer
225 views

A question on relative equivariant cohomology

Suppose that we have defined an extraordinary $G$-equivariant cohomology theory $H$ (say $G$ is a compact group). If $X$ is a $G$-space and $A\subset X$ is a closed $G$-equivarant contractible ...
user avatar
  • 55
2 votes
0 answers
97 views

Computation of equivariant 3 form

I want to how an equivariant 2-form and equivariant 3- form look like i,e., Let M be a complex Manifold say $ S^4 \subset C^3$ and a compact lie group $S^1$ acting on it via the action $\exp{i\theta}....
user avatar
7 votes
1 answer
302 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 ...
user avatar
  • 637
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 ...
user avatar
  • 321
6 votes
0 answers
159 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 ...
user avatar
  • 61
4 votes
0 answers
77 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 \...
user avatar
3 votes
1 answer
253 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 ...
user avatar
6 votes
0 answers
204 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 ...
user avatar
11 votes
1 answer
419 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 ...
user avatar
7 votes
0 answers
335 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,\...
user avatar
5 votes
1 answer
241 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}, \...
user avatar
  • 1,157