# 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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### 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: ...
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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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### 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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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 $\mathbb{C}^2$: $S^3=\{(z_0,z_1)\... 1answer 66 views ### Showing that this map descends to the quotient in an injective way Let$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{...
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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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### 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 $\mathbb{S}^n$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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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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### 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 $$S^2\to \mathbb{C}P^{2n+2}\to\mathbb{H}P^n?$$ ...
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### 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 ...
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### 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 ...
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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 ...
### 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 ...