# Questions tagged [equivariant-maps]

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

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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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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 $G\times X\to Y$, $(g,x)\mapsto F(... 2 votes 1 answer 130 views ### G-equivariant automorphisms of G Let$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{... 57 views

### 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
227 views

### 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 ...
1 vote
176 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 ...
91 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 ...
161 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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### 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 (... 19 views

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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) ...
345 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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### 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, ...
20k 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 ...
809 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 ...
101 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
142 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
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. ...