# 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 $(R \otimes R/C_2) \neq 0$ ...
1 vote
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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 ...
1 vote
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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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### 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 $G$-...
1 vote
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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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### 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 $G$, 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 $f:X\... 1 vote 1 answer 73 views ### G-equivariance at the Lie algebra level Let$\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 ... 0 votes 0 answers 35 views ### 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$\mathbb{R}^{d} \rightarrow \mathbb{R}^{k}$, ... 3 votes 1 answer 57 views ### 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 ... 2 votes 0 answers 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. ... 2 votes 0 answers 57 views ### 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: $${\cal{L}}^G\;=\;\left\{~\varphi~\Big{|}~~~\varphi:\,~G\... 3 votes 1 answer 121 views ### Equivariant Diffeomorphism S^2 \times S^3 to itself with respect to the following \mathbb{Z}_4 action Let \mathbb{Z}_4 be the cyclic group generated by (R,j) where R \in SO(3) is the rotation matrix R = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{... 2 votes 0 answers 51 views ### Strange notation of fibre product in an article of Cattaneo, Felder and Tomassini. I'm doing my bachelor thesis on explaining an article of Cattaneo, Felder and Tomassini, and here is the link to the article: https://arxiv.org/pdf/math/0012228.pdf. I'm currently working on the page ... 0 votes 1 answer 112 views ### Equivariant rank theorem. I was reading this post that explains why$$\phi :O_{n}\times H\rightarrow GL_{n}(\mathbb{R})\phi(B,A)=BA$$is a diffeomorphism. Here O_{n} is the orthogonal group and H is the group of all ... 2 votes 1 answer 148 views ### Pushforward of equivariant sheaf I work over an algebraically closed field of characteristic zero. Let G be an algebraic group, X,Y varieties with G-actions, and \phi:X\to Y a G-equivariant morphism. Let \mathcal{F} be ... 3 votes 0 answers 29 views ### Why is the Tate map lax monoidal (just for abelian groups?) Let A be an abelian group with G action, then I call the tate map$$(-)^{tG}: Ab^G \rightarrow Ab A \mapsto \operatorname{coker}(Nm:A_G \rightarrow A^G)$$where Nm:x \mapsto \sum_g gx, is ... 2 votes 1 answer 161 views ### Understanding implications of the equivariant Darboux-Weinstein theorem I am trying to understand the implications of the equivariant Darboux-Weinstein theorem, stated here: The book that states this (Hamiltonian Group Actions and Equivariant Cohomology) gives an example ... 0 votes 1 answer 55 views ### Complex group representations as an enriched category? In my lecture notes it says: ‘Complex representations of a given group G, together with intertwiners, form a category enriched over the complex numbers.’ Is it true that the category is enriched ... 0 votes 1 answer 55 views ### long exact sequence of representations Let G be a group and V_1,V_2 be G-representations. Are the {\rm Ext}^i(V_1,V_2) G-representations as well? Once established this, suppose I have a short exact sequence$$0\to V_1\to V_2\... 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 (... 0 votes 1 answer 18 views ### 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}=... 1 vote
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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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1 vote
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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 ...
1 vote
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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 ...
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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 $i \... 4 votes 1 answer 391 views ### Is the image of an equivariant map always a weakly embedded submanifold? Let$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$be a smooth, ... 4 votes 5 answers 10k views ### Satisfying explanation of Aristotle's Wheel Paradox. The paradox: We have a circle and there is another circle with smaller radius. They are co-centeric. If circle make full turn without sliding, both smaller and bigger circle make full turn too. If ... 33 votes 1 answer 761 views ### Are there intersection theoretic proofs for Ham-Sandwich type theorems? Concrete Question: Let$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$is multiplication of co-ordinates. Can ... 2 votes 1 answer 94 views ### Existence of a certain equivariant map from the sphere to a compact Lie group of lower dimension. Is there a natural number$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}$? If yes, is there a map with this ... 1 vote 1 answer 100 views ### Criterion for equivariant maps to be fiber bundles? Suppose that$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$... 1 vote 0 answers 154 views ### Does the fixed point functor preserve colimit and limit? Let$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)-$spaces. ... 4 votes 3 answers 662 views ### What are some interesting functions that are equivariant under rotations in SO(3)? I'm interested in machine learning on 3D point clouds. Are there any interesting functions that are equivariant under rotations in SO(3)? The PointNet paper: https://arxiv.org/abs/1612.00593 already ... 0 votes 1 answer 162 views ###$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} ...
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 ...