# Questions tagged [hopf-fibration]

For questions on Hopf fibrations

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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 $z=\frac{... 1answer 82 views ### What does the surface |zā|² = |zā|² look like? In a quaternionic plane there are 2 axes and each point corresponds to$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$into two ... 2answers 62 views ### Seeing$\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 \... 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 ... 0answers 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)). ... 0answers 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 ... 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 ... 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 ... 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 \... 0answers 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 ... 0answers 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 (... 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 ... 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=... 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}$$where r is a unit quaternion and P_0=(1,0,0) is a fixed point. If a point r \in S^3 is sent by the Hopf ... 1answer 248 views ### Inverse Hopf Map I have been playing around with the Hopf map and projections and have a question about the inverse map. The hopf map is defined as \pi: \mathbb{S^3} \mapsto \mathbb{S^2} or$$\pi: r \mapsto ri\bar{... 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 ... 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 ... 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 ... 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? 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 ... 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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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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### 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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### 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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### 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 ...