# Questions tagged [equivariant]

The equivariant tag has no usage guidance.

87
questions

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### G-sheaves on spaces with a free G-action

Let $X$ be a topological space
equipped with an action of a group $G$. In the Tohoku paper, Grothendieck defined
"$G$-sheaves" on $X$ as sheaves equipped with $G$-action on the etale space,
...

4
votes

0
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### equivariant Steenrod algebra

From Hu-Kriz "Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence" we know about calculation of $C_2$ - equivariant Steenrod algebra.
Where can I find (if it ...

4
votes

3
answers

592
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### 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 ...

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0
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26
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### On the higher-dimensional Berry-Robbins problem

Let $C_n(\mathbb{R}^d)$ denote the configuration space of $n$ distinct points in $\mathbb{R}^d$, say $\mathbf{x}_1, \ldots, \mathbf{x}_n$. The symmetric group $\Sigma_n$ acts on $C_n(\mathbb{R}^d)$ ...

3
votes

1
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### 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 ...

4
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### Pushforward of invariant measures (equivariant Moser theorem)

There is a well-known theorem that between any two absolutely continuous Borel probability measures $\mu$ and $\nu$ on $\mathbb{R}^n$ there is an increasing triangular
transformation $T : \mathbb{R}^n ...

5
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0
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108
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### Which operations commute with fractional translation?

Let $\mathbf{v}$ be a real signal (i.e., an infinitely long vector).
A translation operation $T_{s}\mathbf{v}$ with integer $s$ is trivially
defined by $\forall i,s:\left(T_{s}\mathbf{v}\right)_{i}=v_{...

2
votes

1
answer

107
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### Approximation of $C^1$-smooth equivariant maps by infinitely smooth ones

Let $M,N$ be smooth closed manifolds acted by a finite group $G$. Let $f\colon M\to N$ be a $C^1$-smooth $G$-equivariant map.
Is it true that for any $\varepsilon>0$ there exists a $C^\infty$-...

4
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0
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176
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### Are there mathematical/physical applications of these Weyl equivariant maps?

Let $G$ be a compact Lie group and $T$ a choice of maximal torus. Denote the corresponding Lie algebras by $\mathfrak{g}$ and $\mathfrak{t}$. Elements of $\mathfrak{t} \otimes \mathbb{R}^3$ are called ...

7
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204
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### Ample divisors on $T$-varieties

Question: how does one use a torus action to help decide whether a divisor or line bundle is ample?
In more detail: I have a (normal, but usually rather singular) $T$-variety $X$, where $T$ is an ...

2
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### Equivariant smooth approximation

Suppose we have a compact manifold $M$ with the action of a compact group $G$. Consider the space of $C^l$ $G$-equivariant diffeomorphisms $\text{Diff}_G^{l}(M)$ with the $C^l$ topology and the space ...

6
votes

1
answer

318
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### 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 $...

4
votes

1
answer

115
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### Explicit example of an equivariant embedding of $U(n)/( U(k) \times U(n-k))$ into a finite dimensional $U(n)$-representation

We know that if $H$ is a closed subgroup of a compact Lie group $G$ one can find a finite dimensional $G$-representation $V$ and an element $v_0 \in V$ such that $\textrm{Stab}(v_0)= H$. This gives a $...

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104
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### Equivariant Morse theory for non-compact Lie groups

Let $G$ be a Lie group acting properly on a smooth manifold $M$. The (non-equivariant) definition of a Morse function does not carry over to equivariant functions $M \rightarrow \mathbb{R}$ (where $\...

5
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91
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### Is there a smooth $W_{G_2}$-equivariant map from the flag manifold of $U(4)$ to that of $G_2$?

The Weyl group $W$ of $G_2$, is a group of order $12$ which is generated by the subgroup of permutations of $e_1$, $e_2$ and $e_3$ and by by the element $\tau$ which maps $(e_1,e_2,e_3)$ to $(-e_1,-...

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206
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### A group action on another group action quotient: how to best describe the resulting structure and does it have a name?

Suppose I have an action $\alpha:G\times X\to X$ of a group $G$ on a set $X$ and, on top of that, an action $\beta:H\times(X/G)\to(X/G)$ of another group $H$ on the set of $G$-orbits.
Is there a nice ...

23
votes

1
answer

694
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### Vector bundles on $\mathbb{A}^n / G$

Let $G$ be a finite group acting linearly on $\mathbb{A}^n$. Do we expect algebraic vector bundles on $X := \mathbb{A}^n/G$ to be trivial? Here by the quotient I mean the singular scheme, not the ...

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0
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55
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### Singular chain complex of balanced products

Let $\pi\subseteq\Sigma_r$ and $V$ be a right $\pi$-space. We may assume that $V$ is free, if necessary. Consider the morphism of singular chain complexes (over a fixed commutative ring)
$$f:C_*(V) \...

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0
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240
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### Definition of an equivariant connection and equivariant curvature

Can anyone give me a reference which precisely stated the definition of an equivariant connection and equivariant curvature?
Precisely, If G be a compact lie group acting linearly on a smooth ...

8
votes

1
answer

425
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### Terminology about G- simplicial complexes

For a simplicial complex $X$ with an action of a discrete group $G$, we can impose the following condition, namely that if $g\in G$ stabilizes a given simplex $\sigma\subseteq X$, then $g:\sigma\to\...

5
votes

1
answer

202
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### Graded commutativity of the $n$th Browder bracket

Let $\mathcal{O}$ be a topological operad and $X$ an algebra over $\mathcal{O}$. Then $H_*(X)$ is an algebra (in the category of $\mathbb{Z}$-graded $R$-modules) over $H_*(\mathcal{O})$. Each $e\in ...

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0
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137
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### Equivariant vector bundles whose quotient map preserves the stabilizer

Let $G$ be a compact Lie group which act on a manifold $M$. We fix this action throughout our question.
Assume that $E\to M$ is a vector bundle which has the potential of admitting ...

2
votes

1
answer

206
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### Resolution by locally free $G$-equivariant sheaves on varieties

I have been reading the section in the beginning of Fourier-Mukai and Nahm Transforms in Geometry and Mathematical Physics by Bartocci et. al., and stumbled across the following sentence (page 26).
...

2
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73
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### Does intermediate extension functor commutes with forgetful functor in equivariant derived category?

The forgetful functor from $D^b_G(X)$ to $D^b(X)$ carries $Perv_G(X)$ to $Perv(X)$ by definition $5.1$ in the book of Bernstein and Lunts. My question is do the following functors, intermediate ...

3
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### Could we have the simplicial definition of equivariant derived category of sheaves with arrow direction inversed?

Let $X$ be a topological space and $G$ be a topological group acting on $X$ from the left. We consider the simplicial space $[G\backslash X]_{\cdot}$ where
$$
[G\backslash X]_n=\underbrace{G\times \...

9
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0
answers

136
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### Does the category of $G$-equivariant sheaves have enough injectives?

The question is related to this one.
Let $k$ be a field and $X$ be a topological space. We consider Sh$(X)$, the category of sheaves of $k$-vector spaces on $X$.
Let $G$ be a topological group which ...

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0
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113
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### The space of Riemannian structures as an orbifold.

Consider a smooth closed manifold $M$. The space of Riemannian metrics is an open cone in the space of sections of some vector bundle. On this space the group of diffeomorphisms of $M$ acts by ...

6
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0
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187
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### Fundamental class in equivariant K-theory

I'm looking for an accessible reference for the definition of the fundamental class in equivariant K-theory.
The setup I'm interested in is the following: suppose $V$ is a vector space equipped with ...

4
votes

0
answers

129
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### Spin equivariance of the Dirac operator-flat case

This question was posed on Math.SE but no one has answered it; it may be suitable for MathOverflow.
Let $D$ be the Dirac operator on $\mathbb{R}^n$ i.e. $D=\sum_{j=1}^nE_j\frac{\partial}{\partial ...

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160
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### Baum Connes conjecture and abstract isomorphism

Baum-Connes conjecture states that for a locally compact group $G$ the so called assebly map $\mu$ between $G$-equivariant K-homology of the universal example for proper actions of $G$ and K-theory of ...

8
votes

1
answer

573
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### 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 ...

5
votes

2
answers

720
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### Classification of (complex algebraic) vector bundles on punctured affine space

The Quillen-Suslin theorem asserts that there are no nontrivial vector bundles over the affine space $\mathbb{A}^{n+1}$, $n\geq 0$.
Let's work over the complex numbers. What can be said about vector ...

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0
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78
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### 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

0
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### Equivariant corner straightening

Equivariant corner straightening is usually mentioned in the literature without further explanation. What would be a reference where this is done (more or less) carefully for compact Lie group actions ...

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159
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### Generalizing approximate $\mathbb{Z}$-equivariance of a simple function

Let $f(x) := x^2 + (1-x^2)x$ and $F(x) := \log \frac{x}{1-x}-\frac{1}{x}$. It can be shown (cf. https://math.stackexchange.com/questions/1865370/) that $F$ is approximately equivariant w/r/t the $\...

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### group actions of $S^3$ on the configuration space of projective plane

Let $\mathbb{R}P^2$ be the lines in $\mathbb{R}^3$ passing through the origin. Let $SO(3)$ act on $\mathbb{R}^3$ canonically. Then $SO(3)$ has an induced action on $\mathbb{R}P^2$. Let $F(\mathbb{R}P^...

2
votes

0
answers

207
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### Moment map of equivariant line bundles

I'm reading Szabo's `Equivariant Cohomology and Localization of Path Integrals'. I've stumbled upon an equation I can't make sense of, in the discussion about $G$-equivariant line bundles on ...

4
votes

1
answer

401
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### Do equivariant morphisms induce representable maps of quotient stacks?

Let $f: X \to Y$ be a $G$-equivariant map between schemes $X$, $Y$ with action of a flat group scheme $G$. Then why is the induced map of algebraic stacks $[X/G] \to [Y/G]$ representable?

3
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### Equivariant Riemann-Roch on DM stacks?

Does an equivariant version of (Toen)-Riemann-Roch theorem hold say over a smooth Deligne-Mumford stack $X$ over the complex numbers?
Any references that state this explicitely?
Are there formulas ...

6
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0
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### 2-functoriality of equivariant derived categories

I am wondering about the 2-functoriality in equivariant derived categories, and I hope that someone can clarify... (apologies if this is a stupid question)
For the more precise formulation, recall ...

5
votes

2
answers

1k
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### Atiyah-Guillemin-Sternberg convexity theorem

I would like to study the Atiyah-Guillemin-Sternberg convexity theorem: proof and applications. I am already familiarised with hamiltonian actions, moment maps...and with elementary Morse theory.
So ...

3
votes

0
answers

324
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### Equivariant form of Nagata's compactification theorem?

Given a finite group $G$ acting on an algebraic variety $X$ (let's say over $\mathbb C$, if that helps), is there always a proper variety $\bar X$ with a $G$ action such that $X \to \bar X$ is a $G$-...

4
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0
answers

202
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### Milnor's model of $EG$ and Kac-Moody groups

I am working with non-compact Kac-Moody groups $\mathcal{K}$. We can use Milnor's join model for $E\mathcal{K}=\varinjlim \mathcal{K}^{*n}$, where $\mathcal{K}^{*n}$ is the iterated join (see page 20 ...

3
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### good choice of extension of equivariant map

Let $K$ be a compact Lie group. Let $C_K$ denote the category whose objects are the compact lie groups containing $K$ and whose morphisms are inclusion of the groups. Let $Y$ be a $K-$space such that $...

2
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1
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### How to extend an equivariant map from a compact Lie group

Let $G$ be a compact Lie group and let $H$ be a closed subgroup of it. Let $g$ be a torsion element of $G$ and $C_G(g)$ the centralizer of it. Let $Y$ be a $C_G(g)-$space. I'm working on the space $$...

3
votes

1
answer

136
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### What's the topology on the mapping space $Map_H(G, Y)$ when $G$ is not finite

When $G$ is a finite group and $H$ a closed subgroup of it, the sets of right cosets $H\backslash G$ has the discrete topology on it. Let $Y$ be a $H-$space. We have the $G-$homeomorphism \begin{...

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1
answer

562
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### Beilinson-Bernstein localization: $\mathfrak{g}$ action on $G$-equivariant sheaf

I have a few elementary questions related to Beilinson-Bernstein localization.
Let $G$ be a semisimple algebraic group over $\mathbb{C}$ with Lie algebra $\mathfrak{g}$. Consider the setup of ...

2
votes

1
answer

579
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### Definition of Givental $J$-function of cotangent bundle of flag variety

I would like to know the definition of Givental $J$-function of cotangent bundle of flag variety. To state my question more precisely, let us briefly recall the definition of the Givental $J$-function ...

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2
answers

2k
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### Equivariant version of Morse theory

Is there some variant on Morse or Morse-Bott theory yielding equivariant (co)homology instead of singular homology?
Any reference/idea would be greatly appreciated.
Crossposted on StackExchange.

1
vote

0
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139
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### some intuition about the degree of a map

Consider a map
$$ f: \Sigma \to X/\sigma,$$
where $\Sigma=\Sigma_g/\Omega$ is a quotient of a Riemann surface by an antiholomorphic involution,
$\sigma:X\to X$ is an antiholomorphic involution
of some ...