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Questions tagged [hopf-fibration]

For questions on Hopf fibrations

5
votes
1answer
948 views

Metric on total space, connection and Hopf fibration

As is well known that the three sphere $S^3$ may be viewed as a $S^1$ bundle over base space $S^2$. And it is more interesting, but likewise confusing to me that the metric on $S^3$ may be written as: ...
13
votes
0answers
136 views

Do Hopf bundles give all relations between these “composition factors”?

Write a fiber bundle $F\to E\to B$ in short as $E=B\ltimes F$ (in analogy with groups). (This is not necessary, but: given another bundle $X\to B\to Y$, we can write $E=(Y\ltimes X)\ltimes F$, but ...
10
votes
0answers
466 views

Visualizing Hopf fibration $S^3\to S^2$ as a based map $S^1\to \mathrm{Map}(S^2,S^2)$

A fiber bundle $F\to E\to B$ may be interpreted as $E$ being a bunch of $F$s arranged in the shape of a $B$. For instance, a Mobius band $M$ is a bunch of line segments $[0,1]$ arranged in the shape ...
3
votes
0answers
192 views

Prove that $\pi:S^3\rightarrow \mathbb{P}^1(\mathbb{C})\cong S^2$ is a submersion

I have to solve this one: Let us consider the sphere $S^3\subset \mathbb{R}^4\cong\mathbb{C}^2$, and the map $\pi:S^3\rightarrow \mathbb{P}^1(\mathbb{C})\cong S^2$ defined as the projection's ...
3
votes
0answers
103 views

On the distance function of $\mathbb{C}P^n$

Let $\mathbb{C}P^n$ denote the complex projective $n$-space, endowed with the metric that makes the quotient map $\pi : \mathbb{S}^{2n+1} \to \mathbb{C}P^n$ be a Riemannian submersion. Given $p_1, ...
3
votes
0answers
56 views

Is the generalized hopf map of an alternative finite-dimensional real division algebra continuous?

Let $A$ be an $n$-dimensional alternative real division algebra (not necessarily associative). Is the map $$ \eta \colon \bigl\{(x,y) \in A \times A : |x|^2+|y|^2=1\bigr\} \to A \sqcup \{\infty\}, \...
2
votes
0answers
122 views

Spherical coordinates on the 2-sphere and k-forms

Consider the 2-sphere $S^2$inside $\mathbb R^3$, and let \begin{align} S:(0,\infty)\times(0,\pi)\times(0,2\pi)&\to\mathbb R^3\setminus\{(0,0,0)\} \\ (r,\phi,\theta)&\mapsto(r\sin\phi\cos\theta,...
2
votes
0answers
39 views

How to construct the “lobed Hopf tori”?

I've done the following construction. I expected to get the Clifford torus or another "Hopf torus", such as the ones we can see here: lobed Hopf tori. Here is my construction. First I take the "...
2
votes
0answers
81 views

Continuous sections $S^2 \rightarrow SO(3)$.

We can think of $\operatorname{SO}(3)$ as a bundle with base $S^2 = \operatorname{SO}(3)/\operatorname{SO}(2)$ and fibers $\operatorname{SO}(2)$. There is a well-defined projection $\pi : \...
2
votes
0answers
139 views

Stereographic projection of Hopf map

The Hopf fibration $h:S^{3}\rightarrow S^{2}$ is given by $h(a,b,c,d)=(a^{2}+b^{2}-c^{2}-d^{2},2(ad+bc),2(bd-ac))$. A stereographic projection is a map $s:S^{3}\backslash (1,0,0,0)\rightarrow \mathbb{...
2
votes
0answers
272 views

Distance function on the complex projective space

Let $\mathbb{S}^{2n+1}$ be the Euclidean round sphere of radius 1 and let $\mathbb{C}P^n$ be the complex projective space endowed with the Fubini-Study metric, obtained as the quotient space of that ...
2
votes
0answers
153 views

Hopf map visualization (animation request)

Let $\phi:D^3\to S^2$ be the composition $D^3\to S^3\to S^2$, the first map being the quotient by the boundary and the second map being the Hopf map. Then: $$f_t:x\mapsto(1-t)x+t\phi(x)$$ is a ...
2
votes
0answers
57 views

Platonic Hopf. Given the vertices of a tetrahedron circumscribed by unit sphere, find the stereographic projection of inverse Hopf fibers

I am trying to find the equations in $\mathbb{R}^3$ for the fibers of the four vertices of the tetrahedron circumscribed by the unit sphere. I want to find $s\circ h^{-1}$, where $s$ is the ...
2
votes
0answers
106 views

Lie Bracket, Hopf Fibration, independence of choice

Let $S^3$ be the standard sphere (with the metric induced from $\mathbb{R}^4$) and $\pi\colon S^3\rightarrow \mathbb{C}P^1$ the hopf fibration. Equip $\mathbb{C}P^1$ with the Fubini-Study metric so ...
2
votes
0answers
108 views

How do we understand and visualize the meridians of a 3-sphere?

The equator of a n-sphere is (n-1)-sphere. Then the equator of a 3-sphere is a 2-sphere. Does this mean the meridian of a 3-sphere is a hemisphere (half 2-sphere)? In contrast, Wikipedia describes the ...
0
votes
0answers
57 views

Hopf fibration is fibration indeed

In my topology course, we defined Hopf fibration as the map $p\colon S^3\to \mathbb{C}P^1$, $(z_0,z_1)\mapsto [z_0:z_1]$, where $S^3$ is considered as the unit ball in $\mathbb{C}^2$: $S^3=\{(z_0,z_1)\...
0
votes
0answers
60 views

Composition of stereographic projection and inverse Hopf map

There are several definitions of the Hopf map $H \colon S^3 \to S^2$. The one I use is $$ H(p) = \begin{pmatrix} 2(p_1 p_3 + p_2 p_4) \\ 2(p_1 p_2 - p_3 p_4) \\ -p_1^2 + p_2^2 + p_3^2 - p_4^2 \\ \end{...
0
votes
0answers
49 views

Is there an analogue of the Hopf map for finite fields?

Geometrically, the Hopf map is a continuous map from $S^3$ onto $S^2$, with fibres $S^1$. It can also be viewed as linking complex with real geometry, in some sense. Indeed, $S^3$ can be thought of ...
0
votes
0answers
121 views

What is the associated bundle of Hopf fibration?

I have learned that the Hopf fibration is a principal $S^1$-bundle over $S^2$. The $S^1$-action on $S^3\subseteq\mathbb{C}^2$ is given by $$w\cdot(z_1,z_2)=(wz_1,wz_2).$$ Now let us consider the ...
0
votes
0answers
140 views

Hopf fibration homeomorphism injectivity

I was wondering if there is a possibility to proof the homeomorphism between the 1-dimensional complex projective space $\mathbb{C}P^1$ and the 2-sphere $S^2$ via the Hopf fibration (I know already ...
0
votes
0answers
177 views

Calculating The Fundamental Group of the Hopf Fibration

I want to calculate the fundamental group of the Hopf fibration (or rather, the fundamental group of the total space of the fibre bundle that is the hopf fibration). I know that $S^3$ is simply ...
0
votes
0answers
138 views

Differentiability of Hopf map

Let $S^{2n+1}$ be the unit sphere considered as a subset of $\mathbb{C}^{n+1}$ and define an action of $S^1 \subset \mathbb{C}$ by usual (complex) scalar multiplication. The quotient space is the well ...