# 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.

27 questions
Filter by
Sorted by
Tagged with
155 views

### 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 ...
87 views

### 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 ...
1 vote
85 views

### 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 ...
1 vote
137 views

### 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 $C \subset Z$ ...
177 views

### 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 ...
216 views

### 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 (...
208 views

### 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 ...
137 views

1 vote
99 views

66 views

1 vote