Questions tagged [equivariant-maps]

Questions about or involving equivariant maps, the natural maps between $G$-sets.

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Can the zero map, with non-trivial domain, be an action map?

Edit: So what I have exactly is a $C_2$-equivariant commutative ring $R$, and I want to understand if $(R \otimes R/C_2) \to 0$ could be considered as an action map where $(R \otimes R/C_2) \neq 0$ ...
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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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24 views

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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Confusion on morphism between algebraic varieties with group action

Let $G$ be an algebraic group. I am trying to understand the following isomorphism in the notes[Ch. 2] What I am confused: How one regard a $G$-module $V$ () as a $G$-variety $V^*$. The second ...
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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{...
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  • 707
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What are these 'partial' reflective invariants and equivariants of a multiary function called?

In the univariable case we say that a function is even if $f(x)=f(-x)$ and odd if $-f(x)=f(-x)$ for all $x$ in some space of interest. In the multiary case we would similarly consider a function to be ...
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2 votes
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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 ...
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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 $\...
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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 votes
2 answers
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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 ...
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If $X$ is a principal homogeneous space for a group $G$, show that for every $x\in X$ there’s a unique $G$-equivariant bijection such that $f(x)=1$

If $X$ is a principal homogeneous space for a group $G$, I’m trying to show that for every $x\in X$ there’s a unique $G$-equivariant bijection $f_x:X\rightarrow G$ such that $f(x)=1$ (a map $f:X\...
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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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Dimension of the space of linear equivariant maps

I am reading the paper Provably Strict Generalisation Benefit for Equivariant Models from the latest ICML conference. Suppose we have a linear map $\mathbb{R}^{d} \rightarrow \mathbb{R}^{k}$, ...
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3 votes
1 answer
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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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2 votes
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105 views

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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2 votes
0 answers
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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\...
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1 answer
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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{...
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0 answers
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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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1 answer
112 views

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 ...
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2 votes
1 answer
148 views

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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3 votes
0 answers
29 views

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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2 votes
1 answer
161 views

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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1 answer
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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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  • 1,937
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1 answer
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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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  • 179
2 votes
1 answer
67 views

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 (...
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1 answer
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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}=...
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1 vote
1 answer
144 views

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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2 votes
1 answer
903 views

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 $\...
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2 votes
1 answer
103 views

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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1 vote
1 answer
225 views

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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1 answer
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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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1 vote
0 answers
31 views

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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1 vote
1 answer
60 views

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 ...
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11 votes
0 answers
235 views

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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4 votes
1 answer
391 views

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, ...
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4 votes
5 answers
10k views

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 votes
1 answer
761 views

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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2 votes
1 answer
94 views

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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1 vote
1 answer
100 views

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 $\phi$ ...
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1 vote
0 answers
154 views

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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4 votes
3 answers
662 views

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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0 votes
1 answer
162 views

$G$-bundle Homotopy equivalence

Letting $EG$ be the total space of some group $G$, and $f: X \to Y$ a $G$-equivariant map with $X$ a free $G-CW$ complex, we can form the bundle $p^{\prime}:EG \times_G (X \times Y) \to EG \times_{G} ...
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0 votes
1 answer
85 views

Equivariant Surjective Map

I got the following problem, and I had no clue to prove it. Could someone help me? The problem is: Let $G$ act transitively on a set $X$. Fix $x_{0} \in X$, let $H$ = Stab($x_0)$, and let $Y$ denote ...
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  • 51
2 votes
0 answers
52 views

$spin(n)$ equivariance of Dirac operator

Let $D$ be the Dirac operator on $\mathbb{R}^n$ i.e. $D=\sum_{j=1}^nE_j\frac{\partial}{\partial x_j}$ where $E_j$ are $2^r \times 2^r$ matrices (where $n=2r$ or $n=2r+1$) satisfying $E_i^2=I, \ \ ...
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  • 4,380
2 votes
0 answers
103 views

Equivariant homotopy vs homotopy of fixed points

Let $G$ be a Hausdorff group and $X, Y$ two $G$-spaces. Assume that $f,g:X\to Y$ are two $G$-maps which are $G$-homotopic. If we consider the fixed point functor $(\cdot)^H$ for a (closed) subgroup $H$...
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1 vote
0 answers
76 views

On equivariant maps

It is well know the notion of Equivariant map . Now, consider the situation: $X$ and $Y$ being $G$- and $H$-sets, resp., and a homomorphism $\varphi: G\to H$. How is called a map $f:X\to Y$ ...
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1 vote
0 answers
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On the preservation of group cohomology in certain quotients

Let $G$ be an arbitrary countable group, $X$ a finite, free, proper $G$-CW complex and $C(G)_*$ the induced free, finite $G$-Chain complex. This means that $C(G)_*$ is a finite chain complex, whose ...
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