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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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I have a closed (compact without boundary) manifold M and a compact Lie group G that acts on it. I want to understand the topology of $M/G$, at least compute its singular homology groups. The action ...
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$\DeclareMathOperator{\Z}{\mathbb{Z}}\DeclareMathOperator{\GL}{GL}\DeclareMathOperator{\O}{O}\DeclareMathOperator{\SL}{SL}$Let $\Gamma_0(2)=\left\{\begin{pmatrix} a & b \\ c & d\end{pmatrix} \...
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I am not a specialist in differential geometry, so I would like to ask the following question for clarification. I have a set of objects $M \subset \Bbb{R}^{3k}$ (made of a collection of $k$ points in ...
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The lemma 1.4.1 in van Mill's Introduction to $\beta \omega$ says that if $X, Y$ are compact $F$-spaces and $f:A\to Y$ is continuous where $A\subseteq X$ is a $P$-set, then $X\cup_f Y$ is an $F$-space ...
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Let $X = (\omega_2+1)_\delta$ be the $G_\delta$-modification of $\omega_2+1$, that is $U\subseteq X$ is open iff $U$ is a $G_\delta$-set of $\omega_2+1$, where the latter has order topology. Let $E = ...
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Let $X,Y$ be algebraic varieties and $G$ be a connected reductive group. Assume that there is a Galois cover $f:X \to Y$ with Galois group $\Gamma$ and $G$ acts on $X,Y$ in a compatible way with $\...
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I am new to algebraic spaces and stacks. My question is the following: Let $X$ be a scheme and $G$ be a group scheme action on $X$. Let $[X/G]$ be the quotient stack. Then when the natural map $\pi: ...
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This question is related to this one. If I have an locally closed, quasi projective scheme $X$ contained in an affine space, and a linearly reductive group $G$ acting freely on $X$, are there examples ...
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Let $X$ be a variety (irreducible, normal) over a field $k$ which is algebraically closed with characteristic $0$. Suppose that $X$ has only one singular point $p\in X$, so $Y:=X\setminus p$ is smooth....
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This question has been motivated by p.165 of this book. As in the cited link above, we consider the following space of paraemtrized piecewise $C^1$ loops \begin{equation} X:= \Bigl\{ x : [0,1] \to \...
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Take the $1$-parameter real analytic Schwartz class $\phi_s(x)=e^{\frac{s}{\log x}}$ for $x\in(0,1)$ and $s>0$ where if $s_1 s_2=1$ then $\phi_{s_1}\sim \phi_{s_2}$ (is glued) s.t. $\phi_{s_1}$ and ...
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What is known about the subgroup $U(n)\subset U(mn)$ for $m,n\in\mathbb{N}$ given by the diagonal embedding $$ \alpha\mapsto \text{diag}(\alpha,\cdots, \alpha),$$ for $\alpha$ appearing $m$ times? For ...
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If $(X,A)$ and $(Y,B)$ are (good) pairs of topological spaces, and $f:X\rightarrow Y$ is a homotopy equivalence such that the restriction $f\restriction_A$ is a homotopy equivalence between $A$ and $B$...
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Consider the standard Cremona involution $i:\mathbb{P}^2\dashrightarrow \mathbb{P}^2$, $[x:y:z]\rightarrow [yz:xz:xy]$. Let $Y$ be the blow-up of $\mathbb{P}^2$ in the three base points of $i$, so ...
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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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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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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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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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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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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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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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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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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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$\DeclareMathOperator\SO{SO}$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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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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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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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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$\DeclareMathOperator\SU{SU}\DeclareMathOperator\PU{PU}$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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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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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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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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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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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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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: $$ d_f(z_1,z_2)\triangleq \inf_{\underset{f(x_i)=z_i}{x_i\in X}} \...
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Let us take $G=\operatorname{Gl}(n,\mathbb{C})$ (considered as a linear algebraic group). Let us take $x \in G$: we know that its orbit $\mathcal{O}_x$ under the conjugation action is isomorphic (as ...
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$\DeclareMathOperator\SO{SO}$ Let $\mathcal{M}$ be an open subset of $\mathbb{R}^n$ endowed with the Euclidean metric and $\mathcal{N}$ be a Riemannian manifold. Assume that $G$ is a Lie subgroup of $\...
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Let $\xi_5$ be a 5-root of the unity. We consider $\mathbb{C}^4/G$, where $G=\left\langle \sigma,\tau\right\rangle$, with $\sigma$ and $\tau$ the automorphisms given, respectively, by the following ...
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Let us define the quotient space: $$ 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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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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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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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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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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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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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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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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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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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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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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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 ...
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Given a commutative, $\mathbb N$-graded ring, one can associate to it a scheme via the $\operatorname{Proj}$ construction. What happens if one tries to copy this procedure but instead of $\mathbb N$ ...
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