Questions tagged [equivariant]
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102 questions
25
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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 ...
- 2,310
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0
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Is the equivariant cohomology an equivariant cohomology?
Suppose a finite group $G$ acts piecewise linearly on a polyhedron $X$. Then there are two kinds of equivariant cohomology (or homology).
$\bullet$ With coefficients in a $\Bbb Z G$-module $M$. A ...
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3
answers
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Formality of classifying spaces
Let $G$ be a compact Lie group (or reductive algebraic group over $\mathbb{C}$), and let $BG$ be its classifying space. Fix a prime $p$. Let $\mathcal{A}$ denote the dg algebra of singular cochains on ...
- 6,162
21
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0
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Can we describe equivariant vector bundles of free group action in terms of descent theory (Barr-Beck theorem)?
It is well known that for a compact topological group $G$ acts (say, from the right) freely on a compact space $X$. Then the category of equivariant complex vector bundles on $X$, $\text{Vect}_G(X)$, ...
- 9,019
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8
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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 ...
18
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2
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What is the role of equivariance in the Atiyah-Singer index theorem?
I'm trying to read through "The index of elliptic operators I," and here's what I understand of the structure of the proof:
Define (using purely K-theoretic means) a homomorphism $K_G(TX) \to R(G)$...
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17
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1
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Geometric interpretation of filtered rings and modules
Let $A$ be a commutative algebra, say over $\mathbb{C}$.
Giving a grading on $A$ corresponds at least morally to giving a $\mathbb{C}^*$ action on spec(A): $A_i$ can be thought of as those ...
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16
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2
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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.
11
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1
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What does this naive attempt at $S^1$-equivariant homology describe?
After reading Cohen and Voronov's notes on string topology, one can find the following construction: Suppose we have a topological space $X$ with continuous action of $S^1$. This means we have a map $\...
- 8,167
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1
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Equivariant resolution of singularities
I am looking for some references on equivariant resolution of singularities. In most references quoted on mathoverflow (for instance : Reference on an equivariant resolution of singularities), they ...
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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 ...
- 9,019
10
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0
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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 ...
- 9,340
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0
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Why is the Nil-Hecke Algebra appearing?
The Nil-Hecke algebra is defined to be the subalgebra of the endomorphism ring of $\mathbb{C}[x_1,\ldots,x_n]$ generated by the operators of multiplication by $x_i$ and the divided difference ...
- 8,866
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3
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Is this a definition of equivariant derived category?
Let $X$ be a topological space and $G$ be a topological group acting on $X$, both locally compact Hausdorff. Denote by $D^b(X)$ the derived category of sheaves (say of abelian groups) on $X$. We ...
- 13.2k
9
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1
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814
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Are these notions of strongly equivariant D-modules equivalent?
It seems that there are two notions of strongly equivaraint $D_X$- Modules and I would like to know if they are equivalent, or at least how they are related.
Let $\rho: G\times X \rightarrow X$ be an ...
- 13.2k
8
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1
answer
757
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Equivariant algebraic K-theory of affine space
Unlike algebraic K-theory, equivariant K-theory of affine space (over a field $k$) can be quite nontrivial, depending on the action of the group in question. For example, if one takes the standard ...
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1
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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\...
- 18.7k
8
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1
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698
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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 ...
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7
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0
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258
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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 ...
- 6,162
6
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6
answers
599
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Equivariant homology of $\Omega X$\/-space (references needed)?
Let $(X, *)$ be pointed a (1-connected) space, and let $\Omega X$ denote its based loops space. Then, as one knows very well, $\Omega X$ is a group up to homotopy (this includes all the necessary ...
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6
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2
answers
812
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Weights on equivariant cohomology?
Let $X$ be a quasi-projective variety over the complex numbers, equipped with an action of a linear algebraic group $G$.
Is there a natural mixed Hodge structure on its equivariant cohomology?
Is ...
- 13.2k
6
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1
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283
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effective descent of coherent sheaves
I am new to stacks and algebraic spaces. I have the following question:
Let $X$ be a scheme and $G$ be a group scheme action on $X$. Then $X/G$ exists as an algebraic space. Let $\pi: X \to X/G$ be ...
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6
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1
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489
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Equivariant handle decompositions
Suppose I have some smooth closed high-dimensional manifold $M$ acted on smoothly by a finite group $G$. By a metric averaging procedure, we can equip $M$ with a smooth Riemannian metric so that $G$ ...
- 18.7k
6
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1
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377
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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 $...
- 18.7k
6
votes
1
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512
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Cocycle condition for equivariant sheaves
Let $G$ be an affine group that acts on a variety $X$. Equivariant sheaves on $X$ could be defined in the following way. Consider the simplicial space $X_\bullet$
: $X_n := G^n \times X$, $s_0:X_0 \...
- 1,545
6
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2
answers
917
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Do fixed point sets in equivariant crepant resolutions have the same cohomology? How about for the specific case of Nakajima quiver varieties?
A crepant resolution $f:Y\to X$ is a resolution of singularities with $f^*(K_X)=K_Y$. Crepant resolutions do not always exist, and when they exist they may not be unique. However, different crepant ...
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6
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0
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213
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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,
...
- 9,237
6
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0
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178
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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 $\...
- 198
6
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0
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245
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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 ...
- 18.4k
6
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0
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237
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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 ...
- 371
6
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0
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114
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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 ...
- 17.4k
5
votes
3
answers
681
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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 ...
- 8,866
5
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2
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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 ...
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5
votes
2
answers
1k
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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 ...
- 23.4k
5
votes
1
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564
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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?
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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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2
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Jets of Equivariant Vector Bundles
Let $M$ be a (compact) $G$-homogeneous space with fibre group $H$, and let ${\cal E}$ be a $G$-equivariant $k$-dimensional vector bundle over $M$ with corresponding representation $\pi:H \to $R$^k$. ...
- 3,409
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0
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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 $G/H\cong S^{2k-1}$ ...
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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 ...
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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_{...
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0
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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,-...
- 5,215
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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 ...
- 29.1k
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0
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334
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T-Equivariant trivialization of a principal G-bundle
Let $k$ be a field, let $G$ be an algebraic group scheme over $k$ and let $T = \textrm{Spec } k[x, x^{-1}]$ be a one-dimensional torus. Does there exist
a scheme $X$ over $k$,
an algebraic $T$-action ...
- 562
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0
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624
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What is the right notion of equivariant Cech cohomology?
What is the right definition of equivariant Cech cohomology is so that given a $G$-space $X$, $H^1_G(X;H)$ classifies $G$-equivariant principal $H$-bundles on $X$?
- 1,138
4
votes
2
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522
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Equivariant K-theory of $S^1$-action on $S^2$
Is there any references for the structure of the equivariant K-theory $K_{S^1}(S^2)$ where the action of $S^1$ on $S^2$ is defined to be rotation about the $z$-axis? What is the ring structore of $K_{...
- 9,019
4
votes
1
answer
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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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1
answer
315
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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 ...
- 11.8k
4
votes
1
answer
182
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Equivariant versus retractive spaces: a reference request
Let $T$ be the category of compactly generated weak Hausdorff spaces with model structure given by Serre fibrations, Serre cofibrations and weak homotopy equivalences. Let $G = |G.|$ be the (geometric)...
- 18.9k
4
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0
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155
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$\pm 1$-equivariant perverse sheaves on the affine line
Let $G=\mathbb{Z}/\mathbb{2Z}$ act by the map $z\mapsto -z$ on a complex line $\mathbb{C}$. The category $\mathcal{Perv}(\mathbb{C})$ of perverse sheaves smooth along the stratification by the origin ...
- 1,071
4
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0
answers
135
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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 ...
- 41