# 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 $(R \otimes R/C_2) \neq 0$ ...
• 21
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 ...
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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 ...
• 1,336
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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 $G$-...
• 59
1 vote
63 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 ...
• 2,110
1 vote
74 views

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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• 4,442
1 vote
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 ...
• 713
1 vote
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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### 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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