# Questions tagged [hopf-fibration]

For questions on Hopf fibrations

49 questions
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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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### 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 ...
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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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### 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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### 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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### 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 ...
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### 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 ...
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 ...
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 ...