# Questions tagged [hopf-fibration]

For questions on Hopf fibrations

49 questions
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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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, ...
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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 $... 2answers 674 views ### Hopf fibration and$\pi_3(\mathbb{S}^2)$I am interested in Hopf's original argument showing that$\pi_3(\mathbb{S}^2)$is non-trivial (using his fibration). It should be exposed in his paper Über die Abbildungen der dreidimensionalen Sphäre ... 1answer 496 views ### Global Section for Hopf Fibration I want to know the existence of global section of$\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$there exists no ... 1answer 475 views ### What is$\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 ...
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 ...