# Questions tagged [quotient-space]

Quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given topological space. The points to be identified are specified by an equivalence relation. This is commonly done in order to construct new spaces from given ones. The quotient topology consists of all sets with an open preimage under the canonical projection map that maps each element to its equivalence class.

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### Isomorphic quotients of a countably infinitely-generated free abelian group

Let $F$ denote the free abelian group on countably infinite generators. I am trying to understand the relationship between normal subgroups $A$ and $B$ of $F$ with isomorphic quotients. So is there a ...

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### How to define the Sobolev quotient space $H^s(Γ)/{\mathbb R}$

Let $\Gamma$ be the boundary of a Lipschitz domain $\Omega\subset \mathbb R^3$. Denote by $H^s(\Gamma)$ the usual scalar Sobolev space for $s\in\mathbb R$. I want to know the definition of the ...

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### Efficiently finding solutions to the Rainbow cryptosystem using quotient spaces

Repost of a mathematics stackexchange question here as this concerns my research and it went unanswered on there.
In this paper, Ward Beullens gives another way to look at the Rainbow cryptosystem. In ...

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### When does base-change in topological spaces preserve quotient maps?

The question when $(-) \times X$ preserves colimits in topological spaces is well-studied. Since it always preserves arbitrary coproducts (disjoint unions), one only has to show when it preserves ...

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### Niceness properties of quotient spaces by continuous equivalence relations

Given an equivalence relation $R$ on a topological space $X$, there are certain conditions we may ask of $R$ that imply certain well-behavedness conditions on the quotient space $X/\mathord{\sim}_R$. ...

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### Can a compact good orbifold be realized as a global quotient of a compact manifold?

Let $\mathcal{O}$ be a compact good orbifold, where we understand a good orbifold to be an orbifold obtained as a global quotient $M/G$, where $M$ is a manifold and $G$ is a discrete group. Are there ...

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### Continuous surjection between spectra of commutative von Neumann algebras

Suppose that $V_1,V_2$ are two commutative von Neumann algebras and $V_1 \subset V_2$. Being in particular commutative $C^*$-algebras we have that $V_1 \cong C(X_1), V_2 \cong C(X_2)$ for some ...

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### Known structures on space of vector flows on manifold

Suppose $M$ be a smooth manifold with some conditions/structures (1). For instance, metric, holomorphic structure, etc..
Then, let $X$ be a nowhere vanishing vector field that respects the (1) of $M$. ...

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### Is this quotient of $\mathbb{C}^{m+1}$ by $U(1)$ only "nice" for $m=1$?

Let $V^{m+1} = \mathbb{C}^{m+1}$ and let $U(1)$ act on it by its diagonal representation, so that really, it is just like scalar multiplication by a unit modulus complex number.
I am interested in the ...

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### Coordinates on quotient manifold $\mathrm{SO}(3)/\Gamma$

$\DeclareMathOperator\SO{SO}$Say I have coordinates for $\SO(3,\mathbb{R})$, e.g., a parametrization by Euler angles. Is there a reasonable way to explicitly prescribe coordinates on the quotient ...

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### Geometric quotients of DM stacks by group actions

Let $G$ be a finite group acting on a DM stack $X$ and $Y$ the quotient stack. I.e., $Y \to BG$ has fiber $X$. Is there a geometric quotient $X/G$ in some sense? I want automorphism groups of points ...

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### Is the restriction of a projection to a compact subset a quotient map?

Let $(X, \mathcal{T}_X)$ and $(Y, \mathcal{T}_Y)$ be topological spaces, $Z = X \times Y$, $\mathcal{T}_Z$ be the product topology on $Z$, $f : Z \to X$ be defined by $f(x, y) = x$, and $C \subset Z$ ...

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### Freely add all quotients to a category

I would like to know if there is some uniform construction out of a given category $\mathcal C$ that freely throws in all quotients,to form a new category $\mathcal C'$. Preferably $\mathcal C'$ has ...

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### CW structure on $\mathrm{PU}(3)$/Heisenberg group

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\PU{PU}$Consider the quotient space $\PU(3)/H=\SU(3)/G_{81}$ where
$H$ is the Heisenberg group of order 27
$G_{81}$ is the No. 9 group of order 81 (...

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### Quotient variety and subgroups

Let $G$ be an affine algebraic group (let's say over $\mathbb{C}$). If necessary one can assume $G$ to be reductive. Imagine one has $X$ over which $G$ acts freely: moreover, we have a locally closed ...

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### Compactness of symmetric power of a compact space

Suppose I have a compact metric space $(X,d)$ and let $\mathcal{X}=X^K$ be the product space. Consider the equivalence relation $\sim$ on $\mathcal{X}$ given as: for $\alpha,\beta\in \mathcal{X}$, $\...

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### Is the composition of group quotients a group quotient?

I have two sets $X_1$, $X_2$ each with a corresponding group action $G_1$, $G_2$. Linking the two sets is $f:X_1\to X_2$ that maps orbits of $G_1$ into the same point in $X_2$. In other words $X_1/...

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### Quotient measure on locally compact spaces

Suppose we are given a locally compact topological space $X$ and a discreet group $G$ acting on it (we can assume the action to be proper). Given a Radon probability measure on the quotient space $G \...

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### Associated fibered space

I am doing research for a university project which consists of studying quotients in algebraic geometry. In some notes concerning principal bundles that given a representation of the structure group $...

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### Condition for: A simple quotient metric induced by surjective map + equivalence relation

Let $X$ be a metric space and let $f:X\rightarrow Z$ be a surjective map onto some set $Z$. Define the pseudo-metric $d_f$ on $Z$ by:
$$
d_f(z_1,z_2)\triangleq \inf_{\underset{f(x_i)=z_i}{x_i\in X}}
\...

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### Geometry of elements with prescribed multiplicity eigenvalues

Let us take $G=\operatorname{Gl}(n,\mathbb{C})$ (considered as a linear algebraic group). Let us take $x \in G$: we know that its orbit $\mathcal{O}_x$ under the conjugation action is isomorphic (as ...

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### Quotient of $\mathbb{R}^n$ by a subgroup of $\mathrm{SO}(n)$

$\DeclareMathOperator\SO{SO}$ Let $\mathcal{M}$ be an open subset of $\mathbb{R}^n$ endowed with the Euclidean metric and $\mathcal{N}$ be a Riemannian manifold. Assume that $G$ is a Lie subgroup of $\...

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### Partial crepant resolution in codimension 2

Let $\xi_5$ be a 5-root of the unity. We consider $\mathbb{C}^4/G$, where $G=\left\langle \sigma,\tau\right\rangle$, with $\sigma$ and $\tau$ the automorphisms given, respectively, by the following ...

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### Is this Beppo-Levi curl space a Banach space?

Let us define the quotient space:
$$ V = \{ \mathbf{u} \in L^2_{loc}(\mathbb{R}^3; \mathbb{R}^3) : \operatorname{curl} \mathbf u \in L^2(\mathbb{R}^3; \mathbb{R}^3) \} / \nabla H^1_{loc}(\mathbb{R}^3)....

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### Description of $A^\bullet(G/H)$ [closed]

Let $G$ be a compact Lie group and let $H$ be a closed subgroup of $G$, with Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$.
We denote $G\times_H \mathfrak{g} / \mathfrak{h}$: the set of orbits $(G \...

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### What is the quotient (pseudo)metric $d_\sim$ and how do I identify the infimum of possible sequences in this instance?

Let $Z$ be the the set of dyadic and ternary rationals in the interval $\left[\frac12,1\right)$ whose 3-adic valuation is either $-1$ or $0$, with the standard absolute value topology inherited from ...

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### Descent of projective bundles

A problem studied in GIT is the descending of vector bundles (or more in general coherent sheaves) to quotients.
It is a result of Kempf that whenever we have a vector bundle over a quasiprojective ...

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### Is the symplectic quotient $\mu^{-1}(0)/G$ unique up to something?

Given a Hamiltonian action of a compact Lie group $G$ on a symplectic manifold $(M,\omega)$, we may choose a moment map $\mu \colon M\to \mathfrak{g}^* $ and obtain the symplectic reduction $M/\!\!/G =...

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### $L_p(I,Y)^\perp=L_q(I,Y^\perp)$?

Let $X$ be a Banach space and $Y$ be a closed subspace of $X$. For $1<p<\infty$ consider the $p$-th power Bochner Integrable functions which takes values in $X$ and defined on the unit interval $...

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### Properness of reductive group actions on smooth varieties

Suppose that $G$ is a reductive algebraic group acting on a smooth variety $X$, and that the action has finite stabilizers. When is the action of $G$ on $X$ proper? What is an example where the action ...

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### When is the quotient of a geodesic space again a geodesic space?

I asked this on the math.stackexchange forum about a week ago but did not get any answers so I figured I might try it here as well. This is a straight up copy paste from my question here.
I am ...

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### Parametrizing quotient of matrices by the orthogonal group

I am trying to parametrize the collection of $d\times m$ real matrices quotient $d\times d$ orthogonal matrices. Formally, define $\sim$ on $\mathbb{R}^{d\times m}$ by $X\sim Y$ if there exists an ...

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### Subspaces of compact spaces and quotients of Hausdorff spaces

Let $\operatorname{Top}$ be the class of topological spaces. Furthermore, let $\mathcal{U}\subset\operatorname{Top}$ and $\mathcal{V}\subset\operatorname{Top}$ classes satisfying the following ...

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### Explanation for "Squashing" and "Stretching" (Lorentzian Analogue of Berger Spheres)

In the paper Anti-de Sitter space, squashed and stretched Bengtsson and Sandin introduce the Lorentzian analogue of the squashed 3-sphere. After looking up Berger spheres, it seems what is meant with "...

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### Quotient of a Fano variety by a torus

We work over an algebraically closed field of characteristic zero. Let $X$ be a Fano variety, and $T\cong \mathbb{G}_m^r$ a torus acting faithfully on $X$.
I think we can canonically linearize the ...