Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [hopf-fibration]

For questions on Hopf fibrations

22
votes
3answers
405 views

Quotient of $S^3\times S^3$ by an action of $S^1$

Consider the action of $S^1$ on the product of 3-spheres $S^3\times S^3$ defined by: $$e^{it}.(z_1, z_2)=(e^{2it}z_1, e^{3it}z_2)$$ where $z_1, z_2\in S^3$. Here we understant $e^{2it}z_1$ as the ...
17
votes
2answers
7k views

What are all these “visualizations” of the 3-sphere?

a 2-sphere is a normal sphere. A 3-sphere is $$ x^2 + y^2 + z^2 + w^2 = 1 $$ My first question is, why isn't the w coordinate just time? I can plot a 4-d sphere in a symbolic math program and ...
14
votes
1answer
477 views

What is $\pi_{31}(S^2)$?

What is $\pi_{31}(S^2)$ - high homotopy group of the 2-sphere ? This question has a physics motivation: 1) There are relations between (2nd and 3rd) Hopf fibrations and (2- and 3-) qbits (quantum ...
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
474 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 ...
9
votes
1answer
615 views

Is the Hopf fibration the only fibration with total space $S^{3}$?

Let $S^{1}$ bundle over $S^{2}$ is homeomorphic (diffeomorphic) to $S^{3}$. Is the Chern class of the fibration 1, i.e. is the Hopf fibration up to isomorphism the only fibre bundle with total space $...
9
votes
1answer
1k views

Understanding the Hopf fibration

I'm taking a class in manifolds, and the Hopf fibration recently came up. I'm trying to get a handle on it, so I'm going to try and explain what I think is going on, and hopefully math.stackexchange ...
7
votes
1answer
187 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\...
5
votes
2answers
684 views

Hopf fibration and $\pi_3(\mathbb{S}^2)$

I am interested in Hopf's original argument showing that $\pi_3(\mathbb{S}^2)$ is non-trivial (using his fibration). It should be exposed in his paper Über die Abbildungen der dreidimensionalen Sphäre ...
5
votes
1answer
107 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 ...
5
votes
1answer
513 views

Pullback of a form using the Hopf fibration

I embed the 2-sphere and the 3-sphere in $\mathbb{R}^3$ and $\mathbb{R}^4$ respectively. Then denote by $\{x_1,x_2,x_3,x_4\}$ the coordinates on the 3-sphere and $\{y_1,y_2,y_3\}$ on the 2-sphere. So $...
5
votes
1answer
950 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: ...
4
votes
2answers
410 views

group actions on spheres

Let $\mathbb{Z}/2$ act on the $m$-sphere $S^m$ freely and properly discontinuously. If the action is not trivial, can we conclude that the action is homotopy equivalent to the antipodal action? That ...
4
votes
1answer
610 views

Showing that the Hopf fibration is a non-trivial fibre bundle

I want to show that the Hopf bundle $$ \mathbb{S}^1 \rightarrow \mathbb{S^3} \rightarrow \mathbb{S}^2$$ is non-trivial as a principal fibre bundle. I have seen hints of several different approaches: ...
4
votes
1answer
290 views

The rotation group $SO(3)$ may be mapped to a $2$-sphere by sending a rotation matrix to its first column. How can I describe the fibres of the map?

The rotation group $SO(3)$ may be mapped to a $2$-sphere by sending a rotation matrix to its first column. How can I describe the fibres of the map?
4
votes
1answer
502 views

Global Section for Hopf Fibration

I want to know the existence of global section of $\pi : M\rightarrow M/G$, where $M$ is a Riemannian manifold with $G$-action. For instance in case of $M=S^2$ and $G={\bf Z}_2$ there exists no ...
4
votes
1answer
132 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^*...
4
votes
1answer
105 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 ...
3
votes
2answers
116 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 ...
3
votes
1answer
243 views

Hopf fibration and exact sequence in homotopy

I have encountered the Hopf fibration $S^1\hookrightarrow S^3\twoheadrightarrow S^2$ when studying smooth principal $G$-bundles. Whenever I google the Hopf fibration, I encounter a remark which boils ...
3
votes
0answers
196 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
1answer
294 views

Seeking a good book or site describing the 3-sphere

would you be able to recommend a good book/chapter or a web site on visualization / structural elements / projection of the 3-sphere. I am trying to locate a good information source on this subject, ...
2
votes
1answer
105 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 ...
2
votes
1answer
202 views

fibration of complex projective space over quaternion projective space

From the fibration $$Sp(1)=S^3\to S^{4n+3}\to \mathbb{H}P^n,$$ can we quotient the action of $U(1)=S^1$ and obtain a well-defined fibration $$ S^2\to \mathbb{C}P^{2n+2}\to\mathbb{H}P^n?$$ ...
2
votes
0answers
123 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
40 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
141 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
274 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
110 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 ...
1
vote
1answer
32 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 ...
1
vote
1answer
231 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_{...
0
votes
1answer
45 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?
0
votes
1answer
36 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 \...
0
votes
1answer
24 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 $\mathbb{S}^n$ ...
0
votes
1answer
61 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 $S^3$?
0
votes
0answers
59 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
1answer
66 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{...
0
votes
0answers
61 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
125 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
178 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 ...