# Questions tagged [hopf-fibration]

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### Hopf fibration extended to bundle over $\mathbb{C}^2$

Consider the Hopf bundle $h:\mathbb{S}^3\rightarrow\mathbb{S}^2$. There is a connection $1$-form $\omega$ oh $h$ which is left $SU(2)$ invariant. In terms of the Euler angles $(\theta,\,\phi,\,\psi)$ ...
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### Explanation for "Squashing" and "Stretching" (Lorentzian Analogue of Berger Spheres)

In the paper Anti-de Sitter space, squashed and stretched Bengtsson and Sandin introduce the Lorentzian analogue of the squashed 3-sphere. After looking up Berger spheres, it seems what is meant with "...
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### Extension of Hopf fiber bundle to (an equivariant) 2 dimensional vector bundle

Let $p:S^3 \to S^2$ be the Hopf fibration which is a result of the standard action of $S^1$ on $S^3$. Is there a $2$ dimensional vector bundle $\tilde{p}:E \to S^2$ such that $S^3\subset E$ ...
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### What is the intuition for higher homotopy groups not vanishing?

The homotopy groups of the spheres $S^n$ (see Wikipedia) vanish for the circle $S^1$ as, naively speaking, there are not higher order holes to be grasped by higher order homotopy groups. This ...
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### Explicit map with Hopf invariant two in any even dimension

It is known that the Hopf invariant for maps from $\mathbb{S}^{4n - 1} \to \mathbb{S}^{2n}$ is nontrivial (and captures the rational homotopy of the spheres). For $n = 1$, the Hopf fibration provides ...
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### Find wrapping angle of helix on a torus

I need some help in calculating the wrapping angle of a spiral helix wrapped on a torus with constant angle against all the meridians of the torus. The wrapping angle (or the angle measured around and/...
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### Does the Hopf construction work for $S^0$?

Given a $0$-connected $H$-space $X$, we can prove that left and right multiplication are (weak) homotopy equivalences. This allows us to construct the Hopf fibrations for $S^1$, $S^3$ and $S^7$. I ...
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### What is known about the Hopf map for quadratic field extensions?

This question is related to my previous post: Is this generalization of the Hopf map for quadratic field extensions surjective? I still would like to know more and, while that post got several votes,...
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I know $\pi_4(S^2)$ is $\mathbb{Z}_2$. However, I don't know how to visualize it. For example, it is well known that $\pi_3(S^2)=\mathbb{Z}$ can be understood by Hopf Fibration. Elements in $\pi_3(S^2)... 0answers 668 views ### How to see the quaternionic hopf map generates the stable 3-stem? I am looking for a direct proof that the quaternionic hopf map generates (after suspension) the 3rd stable homotopy group of spheres. There are some related MO questions, for example: third-stable-... 0answers 568 views ### "Homogeneity" of the Hopf fibration$S^7\to S^{15}\to S^8$[closed] My question has to do with an apparent contradiction I get regarding the Hopf fibration$S^7\to S^{15}\to S^8$. Namely, the two following statements cannot be true at the same time (but I do not see ... 2answers 522 views ### Hopf Tori in$S^3$By means of the Hopf fibration$\pi: S^3 \rightarrow S^2$, U. Pinkall showed that every compact Riemannian surface of genus one can be conformally embedded in$S^3$. More precisely: Let$p$be a ... 1answer 375 views ### The ring of integers looks like the 3-dimensional sphere viewed as the Hopf fibration This question is based on the following phrase: "In a sense,$\textrm{Spec} \ \mathbf{Z}$looks topologically like a 3-dimensional sphere viewed as the Hopf fibration over$\mathbf{S}^2$." See page ... 1answer 686 views ### Hopf fibration inside the retraction of R^4 minus line -> S^2? This was inspired by this question. Let$Y = {\mathbb R}^4 \setminus$a coordinate line, which retracts to${\mathbb R}^3 \setminus$a point, which retracts to$S^2$. What is an explicit immersion$S^...
Problem. How to partition R^3 into pairwise non-parallel lines? A possible solution is to stack infinitely many concentric'' hyperboloids, by increasing radius and decreasing slope. And don't forget ...