All Questions
Tagged with quotient-spaces projective-space
44 questions
2
votes
0
answers
57
views
Isotropy subgroup of $\operatorname{GL}_{n+1}(\mathbb R)$ acting on $\mathbb R \mathbb P^n$
The general linear group $\operatorname{GL}_{n+1}(\mathbb R)$ (as a Lie group) acts smoothly on the real projective space $\mathbb R \mathbb P^n$, via $A \cdot [x] := [Ax]$. By $[x]$ here, I mean the $...
- 229
2
votes
0
answers
45
views
Different topologies for the real projective base as a manifold
I am stuck with the following problem.
I am given the $d$-dimensional real projective space $\mathbb{R}P^d$ as the set of equivalence classes of lines in $\mathbb{R}^{d+1}$, i.e.
\begin{equation}
\...
- 101
1
vote
0
answers
167
views
Quotient of projective group scheme by a finite group action
Let $X$ be a projective group scheme, over some base $S$, and let $G$ be a finite group acting on $X$ by $S$-isomorphisms.
I would like to understand if/when the quotient $X/G$ is representable by a ...
- 1,280
1
vote
1
answer
251
views
Understanding the CW decomposition of the real projective $n$-space
I'm trying to work on proving the following statement:
Let $\Bbb P^n$ be the $n$-dimensional (real) projective space. Then $\Bbb P^n$ has a CW decomposition with one cell in each dimension $0,...,n$, ...
- 1,286
1
vote
0
answers
52
views
Showing that projective spaces are Hausdorff
We touched upon the spaces $\mathbb RP^n$ in our topology class. We were asked to think on whether $\mathbb RP^n$ was Hausdorff.
After a bit of contemplation, I came to conclude that the answer is ...
- 4,540
0
votes
1
answer
51
views
Provee that $\mathbb{P}^2(\mathbb{R})/H \cong S^2$, where $H\subset \mathbb{P}^2(\mathbb{R})$ is a projective line
$\mathbb{P}^2(\mathbb{R})/H$ identifies the topological quotient space given by the relation $x \sim y \Leftrightarrow x=y$ or $x,y \in H$.
I tried to build an identification $\mathbb{P}^2(\mathbb{R}) ...
- 1,210
3
votes
0
answers
56
views
$P(a,b,c)=P(bc,ca,ab)$ weighted projective planes for pairwise coprime $a,b,c$
Let $a,b,c\geq 2$ be pairwise coprime integers. The (complex) weighted projective plane $P(a,b,c)$ is the quotient of $\Bbb C^3-\{0\}$ by the action of $\Bbb C^*=\Bbb C-\{0\}$ given by $t\cdot (x,y,z)=...
- 1,682
1
vote
1
answer
74
views
Let $\pi$ be the quotient map $S^n\to\mathbb R\mathrm{P}^n$. Is $\pi_{\star p}:T_pS^n\to T_{\pi(p)}\mathbb R\mathrm P^n$ an isomorphism?
I am trying to prove the following statement:
Let $\pi$ be the quotient map $S^n\to\mathbb R\mathrm{P}^n$. Then $\pi_{\star p}: T_pS^n\to T_{\pi(p)}\mathbb R\mathrm P^n$ an isomorphism
Here is my ...
- 891
0
votes
0
answers
45
views
What can we get via taking quotient of $\mathbb{C}P^1$ by a finite abelian group?
Let $G_n = \langle\tau\rangle$ and $G_m = \langle\sigma\rangle$ be groups of $n$-th and $m$-th roots of unity. Define action of $G_n \times G_m$ on $\mathbb{C}P^1$ as follows. $$ (\tau^k, \sigma^t) \...
- 2,678
3
votes
1
answer
228
views
Proving ${\mathbb{P}}^n$ is Hausdorff
I am trying to understand and complete the proof that the real projective space ${\mathbb{P}}^n$ is Hausdorff.In my notes it is modeled as${\mathbb{R}}^{n+1}\setminus \{0\}/\sim $ and it goes like ...
- 3,518
6
votes
1
answer
225
views
Weighted projective plane as a quotient of $\Bbb CP^2$
For positive integers $a_0,\dots,a_n$, consider the weighted projective space $\Bbb C\Bbb P(a_0,a_1,\dots,a_n)$, which is the quotient of $\Bbb C^{n+1}-\{0\}$ by the action of $\Bbb C^*=\Bbb C-\{0\}$ ...
- 1,920
1
vote
0
answers
769
views
Homeomorphism between the real projective plane and a quotient of unit square
A bit of context: I'm starting to learn topology using Topology: A Categorical Approach and the quotient topology is defined as being the finest topology on a set $S$ for which the surjective map $\pi ...
- 105
4
votes
1
answer
400
views
Acting on CP(2) by conjugation
In the paper [2], the author defines the group $G$ to be generated by homeomorphisms of ${\bf S}^2\times{\bf S}^2$ that swap coordinates and/or map them to their antipodal point: $$G=\langle(y,x),(-x,...
- 1,359
2
votes
0
answers
71
views
Show that if $n$ is odd, then $\frac {RP^n}{RP^{n-2}} \simeq S^n \vee S^{n-1} $
$\mathbf {The \ Problem \ is}:$ Show that if n is odd, then $X =\frac {RP^n}{RP^{n-2}}$$\simeq S^n \vee S^{n-1} =Y.$
$\mathbf {My \ approach}:$ We know
$RP^n =\frac {S^n}{C_2},$ then is it true that $ ...
- 2,255
2
votes
1
answer
355
views
Obtaining $\mathbb CP^n$ as an identification space of $D^{2n}.$
Prove that $\mathbb C P^n$ is the identification space $D^{2n}/\sim$ where $x \sim y$ if either $x = y$ or $q(x) = q(y),$ where $q$ is the usual quotient map $S^{2n+1} \xrightarrow {q} \mathbb CP^n.$
...
- 2,932
1
vote
2
answers
201
views
Intuition behind equivalence of two identifications obtained from Möbius band.
Consider the real projective plane $\Bbb RP^2$ which can be realized as a quotient space obtained from a square by identifying points on each pair of it's opposite edges in reverse order. It has been ...
- 2,932
1
vote
1
answer
86
views
Proving a result concerning quotient spaces.
Let $M$ be the Möbius strip and $C$ it's boundary circle. Prove that $M/C$ is homeomorphic to $\Bbb RP^2.$
I know that $\Bbb RP^2 \approx D^2/x \sim -x, x \in \partial D^2.$ So if we can able to get ...
- 2,932
0
votes
1
answer
101
views
On a homeomorphism problem of complex projective space.
Show that $\Bbb C P^n - \Bbb C P^{n-1} \cong \Bbb C^n.$
There is an obvious embedding of $\Bbb C P^{n-1}$ into $\Bbb C P^n$ given by $$[z_0,z_1, \cdots, z_{n-1}] \longmapsto [z_0,z_1, \cdots, z_{n-1},...
- 2,932
2
votes
2
answers
160
views
Showing that $(\Bbb C^{n+1} - 0) {/} (\Bbb C - 0) \cong \Bbb C P^n.$
Show that the complex projective space $\Bbb C P^n \cong \Bbb C^{n+1} - 0 / \Bbb C - 0.$
$\textbf {My thought} :$ We know that $\Bbb C P^n : = \Bbb S^{2n+1}/\Bbb S^1,$ where $\Bbb S^1$ is the unit ...
- 2,932
5
votes
1
answer
269
views
When is the restriction of a quotient map $p : X \to Y$ to a retract of $X$ again a quotient map?
Quite a number of questions in this forum deal with the problem when the restriction $p' : A \stackrel{p}{\to} p(A)$ of a quotient map $p : X \to Y$ to a subspace $A \subset X$ is again a quotient map....
- 81.9k
2
votes
1
answer
83
views
The fibers of the quotient map $q:\Bbb A^{n+1}\setminus\{0\}\to \Bbb P^{n}$ are irreducible.
Let $k$ be an algebraically closed field, let $\Bbb A^{n+1}$ be the
$(n+1)$-dimensional affine space, let $\Bbb P^{n}$ be the $n$-dimensional projective space, and let $q:\Bbb A^{n+1}\setminus\{0\}\to\...
- 1,575
0
votes
1
answer
96
views
Restriction of quotient map $\pi: \mathbb{C}^{n+1}\setminus \{0\} \to \mathbb{P}^n (\mathbb{C})$ to $S^{2n+1}$ is closed
My lecturer, to demonstrate that the projective complex space $\mathbb{P}^n(\mathbb{C})$ is Hausdorff, proves that the restriction of the quotient map to $S^{2n+1}$ is a closed map. Here is how the ...
3
votes
1
answer
121
views
Defining topology on projective space via topology of a field
Given a topology on a field $K$ (in my case a non-archimedean local field with valuation $\nu$ and ring of integers $R$ with maximal Ideal $\mathfrak{m}$), the goal is to define a topology on $\mathbb{...
- 33
8
votes
1
answer
406
views
Construction of $\mathbb{CP}^2$ from $S^2\times S^2$
Show that $\mathbb{CP}^2$ is homeomorphic to the orbit space of $(S^2\times S^2)/\mathbb{Z}_2$. The action is given by $(x,y) \mapsto (y,x)$.
My attempt: I can see that $S^2\times S^2$ is a subset of $...
- 698
1
vote
1
answer
64
views
Projection onto projective space is injective iff $\Bbb K=\Bbb F_2$
Consider $\Bbb K^{n+1}$ a vector space over a field $\Bbb K$ and $\Bbb P^n(\Bbb K)\colon\!=\dfrac{\Bbb K^{n+1}\setminus\{0\}}{\sim}$, where $\Bbb \sim $ is defined by: $v\sim w\Leftrightarrow \exists\,...
- 69
1
vote
1
answer
202
views
Question about the map $S^1\to S^1$ in the context of the real projective plane $\mathbb{R}P^2$
I was recently working on an exercise to compute $H^*(\mathbb{R}P^2\times \mathbb{R}P^2,\mathbb{Z})$ and there was a particular step in the solution for which i would like to get a better intuition.
...
- 2,486
2
votes
1
answer
152
views
Embedding Euclidean Space Into Real Projective Space
I'm struggling with what I think should be a pretty straightforward proof.
Let $g:\mathbb{R}^n\rightarrow\mathbb{R}P^n$ be the map defined by $g=p\circ f$ where $p$ is the quotient map $p:\mathbb{R}^{...
- 699
2
votes
2
answers
270
views
Homogeneous Coordinates on a Real Projective Space
I attempt to understand the construction of the standard $C^{\infty}$ atlas on a real projective space from Loring Tu's An Introduction to Manifolds (Second Edition, page no. 79). Tu denotes the ...
- 831
1
vote
1
answer
247
views
Explicit quotient map from $D^n$ with antipodal boundary points identified to $\mathbb{P}^n$
Let $$\mathbb{P}^n = \{l \ \mid \ l \text{ is a line through the origin in } \mathbb{R}^{n+1} \}$$ denote the $n$-dimensional real projective space. What is an explicit quotient map from $D^n$ to $\...
- 13
4
votes
0
answers
1k
views
The symmetry group / isometry group of the complex projective space
question: For the complex projective space of $n$-complex dimensions,
$$\mathbb{P}^n,$$
what is the symmetry group / isometry group of this complex projective space $\mathbb{P}^n$?
Attempt: Naively, ...
- 3,787
2
votes
1
answer
487
views
Projective Space v.s. Quotient Space v.s. Fibration
What are the more precise relations between (a) projective space, (b) quotient space and (c) the base manifold under certain fibration?
(1) Can every projective space (e.g. $\mathbb{RP}^n$, $\...
- 6,049
2
votes
0
answers
205
views
Homogeneous space and quotient space for real projective spaces
We know that
$$
O(n+1)/O(n) \simeq SO(n+1)/SO(n) \simeq S^n,
$$
based on the result of homogeneous space.
Also
$$
PO(n+1)/PO(n) \simeq P^n,
$$
$P^n$ is the projective space.
These are in some ...
- 6,049
1
vote
1
answer
467
views
Homogeneous space and quotient space for complex projective spaces
We know that
$$
O(n+1)/O(n) \simeq SO(n+1)/SO(n) \simeq S^n,
$$
based on the result of homogeneous space.
Also
$$
PO(n+1)/PO(n) \simeq P^n,
$$
$P^n$ is the projective space.
These are in some ...
- 6,049
3
votes
1
answer
1k
views
Canonical projection is open - Projective Space
Show that canonical projection $\pi:\mathbb{R}^{n+1}\setminus \{0\}\rightarrow \mathbb{RP}^{n}$ is open.
I tried to prove it, but I have a hard time pulling the aberts. I search the internet but only ...
- 816
1
vote
1
answer
34
views
Properties of a function on $\mathbb{R}P(2)$
Define a map $F:\mathbb{R}P(2)\to \mathbb{R^3}$ such that $$F([x,y,z])=\frac{(yz,xz,xy)}{{x^2+y^2+z^2}}$$ Clearly, $F$ is well defined since $F([x,y,z])=F([\lambda x,\lambda y,\lambda z])$ where $\...
- 285
0
votes
1
answer
55
views
Explain how $\mathbb{P}^n$ can be seen as a quotient of $\mathbb{R}^{n+1}\setminus\{0\}$ modulo an equivalence that is to be specified.
For my introductionary class in topology I have to do the following problem:
Explain how $\mathbb{P}^n$ can be seen as a quotient of $\mathbb{R}^{n+1}\setminus\{0\}$ modulo an equivalence relation ...
- 1,628
-1
votes
1
answer
144
views
Equivalence relation $x \sim y$ if $x = \lambda y$ for a non-zero real number $\lambda$
Let $X = \mathbb{R}^{n}\setminus \{0\}$, where $0$ is the origin in $\mathbb{R}^n$.
For $x, y \in X $ we define $x \sim y$ as follows:
$x \sim y$ if there exists a nonzero real number $λ$ such that $...
- 35
0
votes
1
answer
471
views
homeomorphic quotient spaces (with the complex projective plane)
For calculations in homology we often need to know how to identify some quotient spaces.
Let $\mathbb{C}P^n:=\mathbb{C}^{n+1}\setminus \{0\} /\sim$, the complex projective space where $x,y\in \...
- 327
2
votes
1
answer
4k
views
How to prove: Real projective plane is a manifold
I need to correct and fill the gaps in my following home work problem
To show $\mathbb{R}P^2$ is a manifold, I defined the map $f:\mathbb{R}P^2\to\mathbb{D}=\{z\in\mathbb{C}:|z|\le1\}$ by $$f([x])=\...
- 18.7k
1
vote
1
answer
265
views
Is this the basis for the topology of the complex projective plane?
I am doing this as a thought exercise to test my understanding of the quotient topology.
Anyway, given an open set $U \subset \mathbb{CP}^2$, if $\pi: \mathbb{C}^3\setminus\{(0,0,0)\} \to \mathbb{CP}...
- 21.6k
0
votes
1
answer
87
views
"Projective" quotient of $\Bbb{Z}^2$
Consider the space of integer points $\Bbb{Z}^2=\{(x,y)|x,y\in\Bbb{Z}\}$.
Consider now the equivalence relation:
$$
(x,y) \sim (x',y') \quad \Leftrightarrow \quad \beta x'=\alpha x,\, \beta y'=\...
- 8,307
4
votes
1
answer
758
views
Struggling to understand real projective space
My ultimate goal is to show that the real projective space $\mathbb{P}^n_{\mathbb{R}}$ is an $n$-manifold. But first I'd like to understand the topological structure of $\mathbb{P}^n_{\mathbb{R}}$.
...
- 2,523
15
votes
2
answers
7k
views
Real Projective Plane is Same as Identifying Antipodal Boundary Points of The $2$-Disc.
$\newcommand{\RP}{\mathbf RP}$
The real projective plane $\RP^2$ is defined as the quotient space $S^2/\sim$, where $\sim$ identifies the antipodal points of $S^2$.
I want to show that $\RP^2$ is ...
- 19k
4
votes
3
answers
596
views
The quotient map $q: \mathbb R^{n+1} \setminus \{0\} \to \mathbb P^n$ is open.
I'd like to show that the quotient map $q: \mathbb R^{n+1} \setminus \{0\} \to \mathbb P^n$ is open, where I'm considering $\mathbb P^n$ as the quotient space of $\mathbb R^{n+1} \setminus \{0\}$ ...
- 2,559