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Questions tagged [hopf-fibration]

For questions on Hopf fibrations

In the mathematical field of topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a $3$-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz Hopf in 1931, it is an influential early example of a fiber bundle. Technically, Hopf found a many-to-one continuous function (or "map") from the $3$-sphere onto the $2$-sphere such that each distinct point of the $2$-sphere comes from a distinct circle of the $3$-sphere (Hopf 1931). Thus the $3$-sphere is composed of fibers, where each fiber is a circle — one for each point of the $2$-sphere.

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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 multiplication by $e^{2it}$ in each component of…
Miguel Moreira
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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 we may also compose $E\to B$ with $B\to Y$ to get…
anon
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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 animate the w parameter, as w goes from .1 to…
bobobobo
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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 the quaternions by $\mathbb H$), the sphere…
Hugo
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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 of a circle $S^1$, so we may write $[0,1]\to M\to…
anon
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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: There are relations between (2nd and 3rd) Hopf fibrations and (2- and 3-) qbits (quantum bits) entanglement, see this reference:…
Trimok
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Understanding the Hopf fibration

I'm taking a class in manifolds, and the Hopf fibration recently came up. I'm trying to get a handle on it, so I'm going to try and explain what I think is going on, and hopefully math.stackexchange will either correct me, or have more…
Sid Raval
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How does the Hopf fibration generalize to maps $S^{2n+1}\to \mathbb{CP}^n$?

I've been reading about Hopf fibrations. In the Wikipedia page, they state that the Hopf construction generalises to higher-dimensional projective spaces. More specifically, they write that The Hopf construction gives circle bundles…
glS
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Is the Hopf fibration the only fibration with total space $S^{3}$?

Let $S^{1}$ bundle over $S^{2}$ is homeomorphic (diffeomorphic) to $S^{3}$. Is the Chern class of the fibration 1, i.e. is the Hopf fibration up to isomorphism the only fibre bundle with total space $S^{3}$?
yess
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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 the Hopf Fibration". I've done a good job getting…
Julianna Bernardi
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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 other. $$p: S^3\to\mathbb{C}P^1,\quad…
SVG
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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: \begin{equation} ds^2 = (d\psi - \cos \theta\…
Kevin Ye
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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 auf die Kugelfläche, but unfortunately I do not…
Seirios
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Techniques for computing the Brouwer degree of a smooth map

This question is relative to John Milnor's Topology from the Differentiable Viewpoint book, more precisely relative to the problems 13,14 & 15 he is giving at the end of his book. Let $\eta:S^3\to S^2$ be the Hopf fibration, with…
alerouxlapierre
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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?
user191141
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