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Questions tagged [equivariant-maps]

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

Let $G$ be a group, and let $X$ and $Y$ be sets with $G$-actions. A map $f : X \to Y$ is said to be equivariant if $f(g\cdot x) = g\cdot f(x)$ for all $g \in G$ and $x \in X$. That is, the following diagram commutes

$$\require{AMScd} \begin{CD} X @>{g\cdot}>> X\\ @V{f}VV @VV{f}V \\ Y @>{g\cdot}>> Y. \end{CD}$$

106 questions
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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 one show that $$0 \neq H^n(\bigcap_{i=1}^{n} X_i)…
Andres Mejia
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0 answers

Quotient space of equivariant vector bundle.

Let $p: P \to B$ be a principal $G$-bundle, and $\pi : E \to P$ a vector bundle with action of $G$ on $E$ such that $G$ acts by vector bundle isomorphisms and $\pi$ is equivariant. Is it always the case that we get a vector bundle $E/G \to B$? If…
unknownymous
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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 $i \geq 0$. Suppose further that there is an action of…
jacob
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7
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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. Let $X$ be a smooth complex variety with…
Michael Mueller
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7
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2 answers

The square map on $SO(n)$

Let $P:SO(n) \to SO(n)$ be the square map $P(A)=A^2$. Does $P$ define a covering map structure? If yes, is there an action of a finite group $G$ on $SO(n)$ such that each fiber of the covering space is a $G$-orbit and each …
Ali Taghavi
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6 answers

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 we assume that the passed road is equal to the…
Micheal Brain Hurts
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6
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2 answers

Reference for principal bundles and related concepts

I am looking for a good reference for fibre bundles on differential manifolds, Ehresmann connections, principal $G$-bundles and principal Ehresmann connections (the $G$-equivariant version of Ehresmann connections). Could anyone advise me on this? I…
Daniel Robert-Nicoud
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5
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1 answer

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(-k)\right\} \\\subseteq \left\{ f: (I^d,\partial…
Mathematics enthusiast
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Why is the space of continuous mappings between two G-spaces a G-space

Let $X$ and $Y$ be $G$-spaces for a group G (you may assume Hausdorff). Let $Y^X$ be the space of continuous mappings from $X$ to $Y$ with the compact open topology. This space carries a conjugation action of $G$. It is often stated that under some…
Eric Schlarmann
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1 answer

Example of CW-complex with $G$-action, which is not $G$-CW-complex

Let $G$ be a quasi-compact, Hausdorff topological group and let $G$ act on a CW-complex $X$ such that the $G$-action sends cells to cells and boundaries of cells to boundaries of cells. Further, assume that if $e$ is an cell of $X$ such that $g\cdot…
Fabio Neugebauer
  • 954
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1 answer

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 $\theta$. Then $\hat{\tau}_n = g(\hat{\theta}_n)$ is…
Joseph Garvin
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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, equivariant, injective immersion. Is $f(M)$ a weakly…
Asaf Shachar
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4
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3 answers

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 found one way to design functions that are…
Andrew Wagner
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0 answers

Do $G$-spaces with equivalent orbit categories also have equivalent fundamental categories?

I have heard it mentioned before that $G$-spaces which have equivalent orbit categories must then have equivalent fundamental categories (sometimes called the equivariant fundamental groupoid). This would then give an analogy to the result that…
Ducky
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4
votes
0 answers

$G$ equivariant quasicoherent sheaves on $X$ as compatible $G$ actions on the total spaces?

Let $G$ be an algebraic group, and $X$ a scheme on which $G$ acts: i.e the $S$ points of $G \times X \to X$ is a group for each affine $S$. Let $F$ be a quasicoherent sheaf on $X$. There is a notion of a $G$ equivariant sheaf here: I am wondering…
Elle Najt
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