Questions tagged [hopf-fibration]

For questions on Hopf fibrations

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37 views

Hopf fibration. Checking local triviality.

I know that similar question were asked and answered but I want to finish this very construction. In other answers different approaches are used (like construction of sections). I would like to know ...
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1answer
300 views

Why is $\pi_7(\mathbb S^4)=\mathbb Z \oplus \mathbb Z_{12}$?

I'm trying to visualize this fact, not prove it. If we consider the (quaternionic) Hopf fibration $p:\mathbb S^7 \to \mathbb S^4$, where $\mathbb S^7$ is the unit sphere in $\mathbb H^2$ (we denote ...
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106 views

A spinorial generalization of the Hopf map

If $V = \mathbb{R}^3$ with the Euclidean inner product $g$, and $S = \mathbb{C}^2$ is the corresponding space of spinors, then there is a quadratic map $h: S \to V^*$, which maps $\psi \in S$ to $h(\...
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1answer
45 views

'polar' forms of quaternions

We call a quaternion $q = q_0 + q_1 i + q_2 j +q_3 k$ purely imaginary if $q_0=0$. Here $q_0,q_1,q_2,q_3$ are real numbers, and $i, j, k$ the three imaginary units. Is there a reference for the fact ...
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1answer
34 views

Strategy for proving a set is the fiber of a (possible) fiber bundle $p: E\to B$

Given a map $p: E\to B$ and some point $b\in B$, what are some ways to show that a set $F$ is the preimage set $p^{-1}(\{b\})$? So far, it seems $F\subseteq p^{-1}(\{b\})$ can be shown by taking ...
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36 views

Hopf fibration is a riemannian submersion

Reference: Peter Petersen, Riemannian Geometry, 3rd edition, Example 1.1.5 Hopf fibration $F: S^3(1) \to S^2 (1/2)$ is defined by $$F(z,w) = \left(\frac{1}{2} (|w|^2 - |z|^2), z\overline{w}\right)$$ ...
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58 views

Can we see that the Hopf map is essential without using Hurewicz?

The Hopf map is the projection of a circle bundle $h\colon S^3 \to \mathbb{CP}^1 \cong S^2$, where the complex numbers of unit length $S^1 \subset \mathbb{C}$ act on the unit vectors $S^3 \subset \...
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1answer
58 views

Fiber bundle/ Hopf's fiber bundle [closed]

The question: A fiber bundle $(E,B=S^2,F=S^1)$ is given. Determine homotopy groups of $E$ in terms of homotopy groups of $S^2$. Can I say-let $E=S^3$ and use Hopf's fiber bundle?
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1answer
67 views

Lie Group structure on $S^2$

I see that people say there is no Lie group structure on $S^2$. But $S^2$ can be identified with $SU(2)/U(1)$ by the Hopf fiberation. Since $U(1)$ is also a normal subgroup in $SU(2)$, can't you ...
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1answer
44 views

Show that $S^2$ and $S^3 \times \mathbb CP^\infty$ have isomorphic homotopy groups.

I want to show that $S^2$ and $S^3 \times \mathbb CP^\infty$ have isomorphic homotopy groups in each degree. My first approach was to calculate the homotopy group of $\mathbb CP^\infty$ and use the ...
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1answer
44 views

Homotopy group of sphere and Hopf invariant

Sejam $h: S^{3} \to S^{2}$ the Hopf fibration, its induced a sequence exact of homotopy groups, given by $$\cdots \to \pi_{n}(S^1) \to \pi_{n}(S^3) \to \pi_{n}(S^2) \to \pi_{n-1}(S^1) \to \cdots $$ ...
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20 views

Showing that an image of a 3D rotation is a Hopf map

I'm studying geometry and am having trouble with an exercise problem. As a disclaimer, the material is in Korean and there might be some inaccurate things I got wrong when I translated them over to ...
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1answer
104 views

How to get latitudinal tori using the inverse Hopf map?

We define the Hopf map as a function from $S^3$ into $S^2$, $f(z_1, z_2) = \frac{z_2}{z_1}$, where $S^3=\{(z_1, z_2) \in \mathbb C^2 | |z_1|^2 + |z_2|^2 = 1\}$ and $S^2$ is the Reimann sphere $z=\frac{...
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1answer
82 views

What does the surface |z₁|² = |z₂|² look like?

In a quaternionic plane there are 2 axes and each point corresponds to $q \in \mathbb C^2$. Now, $|z_1|^2 = |z_2|^2$ should define a surface that divides a 3-sphere $|z_1|^2 + |z_2|^2 = 1$ into two ...
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2answers
62 views

Seeing $\mathbb{S}^3$ as a pullback

$\require{AMScd}$ Using the Hopf Fibration $$ \mathbb{S}^1 \hookrightarrow \mathbb{S}^3 \rightarrow \mathbb{S}^2 $$ and the fibration $$\mathbb{S}^1 \hookrightarrow \mathbb{S}^\infty \rightarrow \...
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2answers
84 views

Why is the curvature of the connection $\bar{z_1}dz_1 + \bar{z_2}dz_2$ on the Hopf fibration not exact?

Let $\pi : S^3 \to S^2$ be the Hopf fibration, where we take $S^3 \subset \mathbb{C}^2$, $S^2 = \mathbb{C}\mathbb{P}^1$, and $\pi(z_1, z_2) = [z_1 : z_2]$. This is a principal $U(1)$-bundle. The form $...
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103 views

Hopf submersion

Let's consider two maps: $H:\mathbb{S}^3\subset \mathbb{C}^2\to \mathbb{CP}^1$ with $H(z_0,z_1)=[z_0:z_1]$ and $h:\mathbb{S}^3\to \mathbb{S^2}$ with $h(x,y,z,t)=(x^2+y^2-z^2-t^2, 2(yz-xt), 2(xz+yt))$. ...
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271 views

Proving Linkedness of Hopf Fibers

So I've been working on understanding the Hopf fibration in terms of quaternions for the past few months, following along with the investigations in David Lyon's paper, "An Elementary Introduction to ...
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1answer
67 views

Simple Characterization of the Hopf Fiberation as Cosets of the Circle Group (Stabilizer of a Point)

A few months, ago before I took my first Algebra class, I asked a naive question about the formula for one of the hopf fibers here A community member gave a really good answer, so I would recommend ...
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2answers
202 views

How does the Hopf map generate $\pi_3(S^2)$?

I have been studying the Hopf fibration which is an example of a map from $S^3$ to $S^2$. It is a member of $\pi_3(S^2)$ and shows that this group is non-trivial. It can be shown using a long exact ...
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1answer
141 views

Using Hopf fibration to calculate $\pi_{3} (S^2)$

The question says: Theorems of Hurewicz and Hopf say that for $k < n, \pi_{k}(S^n)=1$ and $\pi_{n}(S^n)\cong \mathbb{Z}$. Assuming this for the moment, use the Hopf fibration $\eta : S^3 \...
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141 views

Fundamental group of $S^3\setminus $ disjoint union of circles

Consider the Hopf fibration $\;\;\;p : S^3\rightarrow \mathbb{C}P^1 : (z_1, z_2)\mapsto [z_1:z_2]$. Let $x_1, ... , x_n$ be distinct points in $\;\mathbb{C}P^1$. I'm trying to find the fundamental ...
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39 views

Tangent Vectors to Fibers of the Hopf Map

So I'm viewing the Hopf map as a map from $\mathbb{C}^{2} \to \mathbb{R} \oplus \mathbb{C}$ by $(z,w) \to \left(\frac{1}{2}(|z|^{2}-|w|^{2}),z\bar{w}\right)$. I've concluded that the fiber including $(...
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1answer
77 views

Alternate characterizations of the Hopf circles in $S^3$

The Hopf fibration is a continuous map $S^3 \to S^2$ whose fibers are all circles on $S^3$. Is every one of these fiber circles a great circle of $S^3$? (The Wikipedia page implies this at one point ...
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1answer
111 views

Circle in $\mathbb{S}^3$ not mapping to a point in $\mathbb{S}^2$ under Hopf map

The Hopf fibration is a mapping $h:\mathbb{S^3} \mapsto\mathbb{S}^2$ defined by $r\mapsto ri\bar{r}$ where $r$ is a unit quaternion in the form $r=a+bi+cj+dk $ where $a,b,c,d \in \mathbb{R}$ and $ijk=...
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1answer
36 views

Derivation of a particular hopf fiber formula (pi rotation)

The hopf map in terms of quaternions is defined as $$h:r\mapsto R_{r}(P_0)=ri\bar{r}$$ where $r$ is a unit quaternion and $P_0=(1,0,0) $ is a fixed point. If a point $r \in S^3$ is sent by the Hopf ...
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1answer
248 views

Inverse Hopf Map

I have been playing around with the Hopf map and projections and have a question about the inverse map. The hopf map is defined as $\pi: \mathbb{S^3} \mapsto \mathbb{S^2}$ or$$\pi: r \mapsto ri\bar{...
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1answer
33 views

A $S^1$ invariant frame on $S^3$

We consider the action $S^1$ on $S^3$ with $\alpha.(z_1,z_2)=(\alpha z_1, \alpha z_2)$ where $\alpha \in S^1,\;(z_1,z_2)\in \mathbb{C}^2 \; \text{with} \; |z_1|^2+|z_2|^2=1$. Are there three ...
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1answer
167 views

Is the Hairy Ball Theorem equivalent to saying that the Hopf Fibration has no global sections?

The Hairy Ball Theorem states that $S^2$ has no nonvanishing tangent vector fields. But if we did have such a field then we could normalise each vector so that it lay on the unit circle of the tangent ...
2
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1answer
191 views

Differential of Hopf's map

Let $$h : \mathbb{C^2} \rightarrow \mathbb{C \times R} $$ $$h(z_1, z_2) = (2z_1z_2^*, |z_1|^2-|z_2|^2)$$ How do you find the differential of $h$ and show it is onto/surjective? I know that I can ...
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1answer
118 views

Nontriviality of the Hopf Fibration

A simple question how to understand why even though locally $S^3$ is homeomorphic to $S^2\times S^1$, how do you see that globally this is not true?
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1answer
106 views

Symmetry group of Hopf fibration

https://en.wikipedia.org/wiki/Hopf_fibration What is the group of transformations $\subset SO(4)$ that sends every fibre circle to another fibre circle? I think the Lie algebra might be generated by ...
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1answer
88 views

Showing that this map descends to the quotient in an injective way

Let $f : \mathbb{S}^3 \to \mathbb{S}^2$ be the map $$ f(z_1,z_2) = (2z_1 \overline{z_2}, \vert z_1 \vert^2 - \vert z_2 \vert^2), $$ where we regard $\mathbb{S}^3 \subset \mathbb{C}^2$ and $\mathbb{...
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1answer
282 views

Understanding the Hopf Link

I am trying to understand why the preimages of two points under the Hopf fibration are linked. I thought that two circles in $\mathbb{C}^n$ are linked iff one circle intersects the convex hull of the ...
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0answers
300 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,...
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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 "...
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2answers
154 views

Understanding terminology of, fibers, clutchings and Hopf.

I have some questions regarding the terminology of fiber bundles as used in section 3 of this paper; http://www.sciencedirect.com/science/article/pii/S0723086907000151 The section starts off by ...
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305 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 ...
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1answer
40 views

$\mathbb{S}^1$-action and octonionic multiplication can be associated

Let $\mathbb{S}^7$ be the unit sphere of $\mathbb{R}^8$, which can be identified with the unit octonions. The circle $\mathbb{S}^1$ naturally acts on $\mathbb{S}^7$ by complex multiplication: $$z \...
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1answer
314 views

Hopf map by complex numbers

I read somewhere that the hopf map can be expressed as $h(z_{1},z_{2})=\frac{z_{1}}{z_{2}}$ where $h:\mathbb{C}^{2}\rightarrow\mathbb{C}\cup\{\infty\}$. I let $z_{1}=a+bi$ and $z_{2}=c+di$ and $h(z_{...
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246 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{...
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1answer
74 views

Is there an analogue of the hopf fibration for the hemisphere off $S^3$?

The Hopf fibration represents the 3-sphere $S^3$ as the circle $S^1$ fibred over the 2-sphere $S^2$. Does a similar construction exist for the hemisphere of $S^3$?
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427 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 ...
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54 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 ...
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1answer
966 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 ...
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1answer
262 views

Techniques for computing the Brouwer degree of a smooth map

This question is relative to John Milnor's Topology from the Differentiable Viewpoint book, more precisely relative to the problems 13,14 & 15 he is giving at the end of his book. Let $\eta:S^3\...
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1answer
25 views

Largest elementary neighbourhood

The real projective space $\mathbb{R}P^n$ can be defined as the quotient space of $\mathbb{S}^n$by the equivalence relation that identifies antipodal points. The largest open set of $\mathbb{S}^n$ ...
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1answer
171 views

How to show that the map $\pi: z\mapsto ziz^*$ is onto $S^2$?

We identify $S^3$ with the unit quaternions and $S^2$ the unit pure quaternions, and the conjugate of $z=a+bi+cj+dk$ is defined as $z^*=a-bi-cj-dk$. Then we consider the map $$\pi:S^3\ni z\mapsto ziz^*...
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112 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, ...
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155 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 ...