Questions tagged [equivariant-maps]
Questions about or involving equivariant maps, the natural maps between $G$-sets.
84
questions
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Transport of group actions via homeomorphisms
Let $G$ be a topological group acting on the topological space $X$ and let $\phi\colon X\xrightarrow{\simeq} Y$ be a homeomorphism between the spaces $X$ and $Y$. Can I induce a $G$-action on $Y$ by $\...
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Equivariant splitting of short exact sequences with a $\mathbb {Z}/2\mathbb{Z}$-action
Let $G = \mathbb{Z}/2\mathbb{Z}$, and consider a short exact sequence of \emph{free} $\mathbb Z$-modules of finite rank endowed with a $G$-action
$$ 0\to \mathbb Z^n \to \mathbb Z^m \to \mathbb Z^k\to ...
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1
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Equivalent definitions of group action, isomorphism of GSet categories
Context:
For group actions on a set, there are two habitual definitions and a bijection between objects from each definition:
A function $G\times A \rightarrow A$ with the usual properties
A group ...
- 31
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Nontrivial Monodromy of the Universal Stiefel Bundle (and $O(n)$-equivariant vector fields on spheres)
Note: I'm not allowed to embed images into my posts yet, so I've linked my diagrams instead.
Throughout, we will make use of the following result.
Fact. For $H$ a Lie subgroup of $G$, there is a ...
1
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2
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Existence of continuous map from $\mathbb{R}^n$ to $\mathbb{R}^{n-1}$ that "respects norms"
Is there a continuous map from $\pi: \mathbb{R}^n \rightarrow \mathbb{R}^{n-1}$ that is $(\sim_n, \sim_{n-1})$-invariant where $\sim_n$ is the equivalence relation of equality up to orthogonal ...
- 461
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Equivariant Loomis-Sikorski-Stone reprsentation theorem
Let $X$ be a set, $\mathcal{B}$ a $\sigma$-algebra on $X$ and $\mu$ a probability measure on $(X,\mathcal{B})$.
Suppose $\mathcal{C}$ is a $\sigma$-subalgebra of $\mathcal{B}$. Then, the Loomis-...
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Calculate homotopy groups of $\mathbb{Z}_2$-equivariant loop spaces of "complex" topological spaces
Let $X$ be a topological space such that complex conjugation is defined (e.g. $\mathbb{C}^n$) and let us define the set of maps $$S_d:= \left\{f: (I^d,\partial I^d)\to (X,x_0)\mid \overline{f(k)} = f(...
3
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1
answer
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Existence of a principal bundle charts compatible with $f$-equivariant reductions
Let $\pi:P\rightarrow M$ and $\pi':P'\rightarrow M$ be principal $G$ and $H$ bundles respectively, and $f:G\rightarrow H$ be Lie group homomorphism. Let $F$ be a principal bundle homomorphism, that is ...
- 2,073
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Equivariant Submersion Theorem
I have a question about the Equivariant Submersion Theorem (Proposition 2.7) in the article Equivariant control data and neighborhood deformation retractions by Markus J. Pflaum and Graeme Wilkin. The ...
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Two group cohomology are isomorphic to each other if they have the same coefficient.
Suppose $M$ and $N$ are $\mathbb{F}[G]$-modules over a field $\mathbb{F}$, and $G$ is a finite group.
Claim: Suppose there is a map $f: M \rightarrow N$ such that $f$ is a $\mathbb{F}$-vector space ...
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Why are transition maps of principal bundles given by multiplications of elements of the group?
It seems to be a common fact that transition maps of principal bundles are given by multiplication by a group element (in this post,for example.)
The setup is this: Consider a $G$-bundle $E \overset{\...
- 2,409
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Difficulty with converting Yoneda's natural isomorphy into a group isomorphism in the proof of Cayley's theorem
$\newcommand{\A}{\mathscr{A}}\newcommand{\Gc}{\mathscr{G}}\newcommand{\G}{\mathcal{G}}\newcommand{\s}{\mathsf{Set}}\newcommand{\op}{^{\mathsf{op}}}\newcommand{\sym}{\mathsf{Sym}}$I am having ...
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Equivariant maps up to homotopy
Let $G$ be a topological group and $X$, $Y$ be $G$-spaces. Suppose there is a continuous map $F:X\to Y$ which is $G$-equivariant up to homotopy, namely the two maps $G\times X\to Y$, $(g,x)\mapsto F(...
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G-equivariant automorphisms of G
Let $G$ be group, and
$$
\operatorname{Aut}_G(G):=\{f\in\operatorname{Aut}G\mid f(xg)=f(x)g,\forall x,g\in G\}
$$
I want to show that in fact $\operatorname{Aut}_G(G)\cong G$.
$G\subset \operatorname{...
user867836
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answer
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Do equivariant dynamics have invariant equilibria?
Let $\phi_t$ be a one-parameter subgroup of diffeomorphisms of a manifold ($\mathbb{R}^n$ for simplicity). In other words, $\varphi$ is a continuous dynamics.
Suppose that $\varphi_t$ is $G$-...
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Equivariant map between two vector spaces definition and formulation.
Reading the following paper (Proposition 2.2)
https://arxiv.org/pdf/1804.10306.pdf
I'm stuck trying to understand the following:
We have a compact group $\Gamma$, $V,U$ two vector spaces carrying a ...
- 1,884
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A question about extending group actions to maps.
It is known that an action $\alpha: G \times M \rightarrow M$ induces for each group element $g \in G$ a transformation on $M$, $T_g : M \rightarrow M.$
Now what is done to extend this action to $\...
- 1,884
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Does every layer of an equivariant composite function have to be equivariant?
For example, given a composite function $f(x)=f_1(f_2(f_3(x)))$, if $f(x)$ is equivariant to the group $G$ (e.g. $SO(3)$), then is it necessary for $f_1$,$f_2$,$f_3$ to be equivariant to $G$? Thanks.
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A question about equivariance to 3D transformations using semi-direct and direct products.
in the paper: https://arxiv.org/pdf/2010.02449.pdf the author consider some classes of functions that are invariant to the action of the group $G= \mathbb{R}^3 \rtimes SO(3) \times S_n$ on $\mathbb{R}^...
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2
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Equivariant map in compact homogeneous space is a diffeomorphism
I don't see why an equivariant $G$-map $f: M \rightarrow M$, where $M$ is a compact homogeneous space, is necessarily a difeomorphism. Any idea?
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Equivariant tubular neighborhood of an exceptional orbit of a circle action
A pseudofree $S^1$-action on a sphere $S^{2k-1}$ is a smooth $S^1$-action which is free except for finitely many exceptional orbits whose isotropy types $\Bbb Z_{a_1},\dots,\Bbb Z_{a_n}$ have pairwise ...
- 2,076
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G-equivariance at the Lie algebra level
Let $\vec{x}, \vec{p} \in \mathbb{R}^3$ be coordinates on 6D phase space.
Consider the lift of the standard action of SO(3):
$$\psi_O(\vec{x}, \vec{p}) = (O\cdot\vec{x}, O\cdot\vec{p})
$$
This action ...
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Orientation-reversing involution on $S^3$, commuting with a circle action? [closed]
Consider $S^1$ as a subset of $\mathbb{R}^2\cong\mathbb{C}$, and $S^3$ as a subset of $\mathbb{R}^4\cong\mathbb{C}^2$, and define an action of $S^1$ on $S^3$ by $z\cdot(w_1,w_2):=(zw_1,zw_2)$. Does ...
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Equivariant line bundle-divisor correspondence?
In reading about equivariant bundles, I've become a bit confused about how the usual line bundle-divisor correspondence $c_1:\text{Pic}(X)\xrightarrow{\cong}A^1(X)$ works in the equivariant setting. ...
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Induced representations built on sections of an associated vector bundle. Questions on notations
Consider a group $\,G\,$, a vector space $\,{\mathbb{V}}\,$, and a space $\,{\cal{L}}^G\,$ of functions $\varphi$ on this group:
$$
{\cal{L}}^G\;=\;\left\{~\varphi~\Big{|}~~~\varphi:\,~G\...
- 1,882
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Equivariant Diffeomorphism $S^2 \times S^3$ to itself with respect to the following $\mathbb{Z}_4$ action
Let $\mathbb{Z}_4$ be the cyclic group generated by $(R,j)$ where $R \in$ SO$(3)$ is the rotation matrix $R =
\begin{pmatrix}
-1 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0
\end{...
- 4,316
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Strange notation of fibre product in an article of Cattaneo, Felder and Tomassini.
I'm doing my bachelor thesis on explaining an article of Cattaneo, Felder and Tomassini, and here is the link to the article: https://arxiv.org/pdf/math/0012228.pdf. I'm currently working on the page ...
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Equivariant rank theorem.
I was reading this post that explains why
$$\phi :O_{n}\times H\rightarrow GL_{n}(\mathbb{R})$$ $$\phi(B,A)=BA$$ is a diffeomorphism. Here $O_{n}$ is the orthogonal group and $H$ is the group of all ...
- 4,076
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1
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Pushforward of equivariant sheaf
I work over an algebraically closed field of characteristic zero. Let $G$ be an algebraic group, $X,Y$ varieties with $G$-actions, and $\phi:X\to Y$ a $G$-equivariant morphism. Let $\mathcal{F}$ be ...
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Why is the Tate map lax monoidal (just for abelian groups?)
Let $A$ be an abelian group with $G$ action, then I call the tate map
$$(-)^{tG}: Ab^G \rightarrow Ab$$
$$ A \mapsto \operatorname{coker}(Nm:A_G \rightarrow A^G)$$
where $Nm:x \mapsto \sum_g gx$, is ...
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1
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Understanding implications of the equivariant Darboux-Weinstein theorem
I am trying to understand the implications of the equivariant Darboux-Weinstein theorem, stated here:
The book that states this (Hamiltonian Group Actions and Equivariant Cohomology) gives an example ...
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Complex group representations as an enriched category?
In my lecture notes it says:
‘Complex representations of a given group G, together with intertwiners, form a category enriched over the complex numbers.’
Is it true that the category is enriched ...
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long exact sequence of representations
Let $G$ be a group and $V_1,V_2 $ be $G$-representations. Are the ${\rm Ext}^i(V_1,V_2)$ $G$-representations as well?
Once established this, suppose I have a short exact sequence $$0\to V_1\to V_2\...
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What does it mean for right equivariant maps to be left translations?
My book is Connections, Curvature, and Characteristic Classes by Loring W. Tu (I'll call this Volume 3), a sequel to both Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott (...
user636532
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For which values of a, b, c, d do $\theta_{a,b}$ and $\theta_{c,d}$ commute?
In Sets and Groups by Green a question 5 from the chapter 3 reads:
Write $\theta_{a,b}$ for the map of the preceding exercise [which is $\theta(x)=ax+b=\theta_{a,b}$]. Prove $\theta_{a,b}\theta_{c,d}=...
user620319
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If pushforward by equivariant map of structure sheaf is structure sheaf and the space of sections isomorphic, are they isomorphic as G-modules?
Apologies for what may very well be a trivial question from a non AG person. Suppose I have a morphism of varieties $f: X\rightarrow Y$, with $Y$ affine, which is equivariant with respect to the ...
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Does equivariance of the MLE require the function be invertible?
My statistics text states this theorem as if it works for any function $g$:
Let $\tau = g(\theta)$ be a function of $\theta$. Let $\hat{\theta}_n$
be the MLE (Maximum Likelihood Estimator) of $\...
- 1,158
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1
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Taking homotopy fixed points preserves fibrations
I'm reading a paper where they have an appendix about homotopy fixed point sets of a $G$-space, and at some point they claim that if $f:X\to Y$ is a $G$-map that is an ordinary (non-equivariant) ...
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What are some good references to learn about equivariant homotopy theory?
What are some good references to learn the foundations of equivariant homotopy theory/algebraic topology, for someone who has a background in basic homotopy theory and a tad more advanced algebraic ...
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Schur's lemma proves equivalent irreducible representations are equal?
Let $\phi_1, \phi_2$ be representations of a compact Lie group $G \to V$ that are equivalent. Suppose furthermore that they are irreducible. It follows that there is an isomorphism, $T$, so that: $T \...
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Killing homology below middle dimension with equivariant surgery
Assume a finite group acts smoothly on a manifold $M$ of dimension $n$. Suppose $a\in H_i(M)$, where $i=1,\ldots,[n/2]$.
Is there a way to kill $a$ with equivariant surgery and keep the same fixed ...
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Is equivariant immersion by parts w.r.t an action with finitely many orbits an immersion?
Let $M,N$ be a smooth manifolds of dimension greater than $2$. Suppose that there is a Lie group $G$ acting on $M,N$, and that $f:M \to N$ is a smooth injective equivariant map.
Suppose further that ...
- 24.9k
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G-equivariant isomorphism inducing isomorphisms on quotients
Suppose that $X$ and $Y$ are smooth quasi-projective varieties over $\mathbb{C}$ with a holomorphic map $f:X \to Y$ inducing isomorphisms $f_* : H_i(X;\mathbb{Q}) \to H_i(Y;\mathbb{Q})$ for all $i \...
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Is the image of an equivariant map always a weakly embedded submanifold?
Let $M,N$ be smooth manifolds, with a smooth $G$-action on them, by some Lie group $G$. Suppose also that $M$ has a finite number of orbits under $G$'s-action.
Let $f:M \to N$ be a smooth, ...
- 24.9k
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6
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Satisfying explanation of Aristotle's Wheel Paradox.
The paradox:
We have a circle and there is another circle with smaller radius. They are co-centeric.
If circle make full turn without sliding, both smaller and bigger circle make full turn too. If ...
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33
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809
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Are there intersection theoretic proofs for Ham-Sandwich type theorems?
Concrete Question:
Let $f:\mathbb S^n \to \mathbb R^n$ be a $\mathbb Z/2$ smooth equivariant map where the action on the sphere is antipodal, and $\mathbb R^n$ is multiplication of co-ordinates.
Can ...
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1
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Existence of a certain equivariant map from the sphere to a compact Lie group of lower dimension.
Is there a natural number $n$, a compact Lie group $G$ of dimension less than $n$ and a nontrivial (non-constant) continuous map $f:S^n \to G$ with $f(-x)=f(x)^{-1}$?
If yes, is there a map with this ...
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Criterion for equivariant maps to be fiber bundles?
Suppose that $X$ and $Y$ are manifolds, with a transitive $G$ action, where $G$ is some Lie group. Suppose that $\phi : X \to Y$ is a surjective submersion, which is also $G$ equivariant.
Is $\phi$ ...
- 20.4k
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0
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182
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Does the fixed point functor preserve colimit and limit?
Let $G$ be a compact Lie group and $g\in G$. Let $C_G(g)$ denote the centralizer of $g$ in $G$.
Consider the functor $X\mapsto X^g$ from the category of $G-$spaces to the category of $C_G(g)-$spaces.
...
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3
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What are some interesting functions that are equivariant under rotations in SO(3)?
I'm interested in machine learning on 3D point clouds. Are there any interesting functions that are equivariant under rotations in SO(3)?
The PointNet paper:
https://arxiv.org/abs/1612.00593
already ...
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