✎ Hopf Bundle [The Physics Travel Guide]

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====== Hopf Bundle ======

Hopf bundles are the correct mathematical tools that we need to describe the physics around a magnetic monopole or around instantons.

<blockquote>
Yep. My advisor said that if you want to convince aliens that there's intelligent life on earth, tell them about 1) π and 2) the Hopf map.

<cite>https://twitter.com/math3ma/status/837075372617302016</cite>
</blockquote>

<blockquote>
It's famous because the map from the total space to the base was the first example of a topologically nontrivial map from a sphere to a sphere of lower dimension. In the lingo of homotopy theory, we say it's the generator of the group π3(S2).

<blockquote>
Hopf’s construction of P : S 3 → S 2 was motivated by his interest in what are called the “higher homotopy groups” of the spheres (see Section 2.5). Although this is not our major concern at the moment, we point out that P was the first example of a continuous map $S^m → S^n$ with $m > n $that is not “nullhomotopic” (Section 2.3). From this it follows that the homotopy group $π3 (S^2 )$ is not trivial and this came as quite a surprize in the 1930’s.

<cite>page 16 in Topology, Geometry and Gauge fields by Naber</cite>
</blockquote>

<blockquote>
"When line bundles are regarded as models for the topological structure underlying the electromagnetic field the Hopf fibration is often called “the magnetic monopole”."

<cite>https://ncatlab.org/nlab/show/Hopf+fibration</cite>
</blockquote>

<note tip>
Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.
</note>
  
<tabbox Student>

<blockquote>
"The Hopf fibration is a kind of projection from the three-sphere to the two-sphere. The two-sphere is the one you're likely to be familiar with—a beach ball is a good example. The two-sphere is formed by all points which are a constant distance from a center point. We write the two-sphere as to indicate that it is 2-dimensional."
<cite>http://nilesjohnson.net/hopf.html</cite>
</blockquote>

* The best explanation and especially how it is related to [[topology:notions:magnetic_monopoles|magnetic monopoles]] can be found in chapter 0 of the book Topology, Geometry and Gauge fields by G. Naber.
  * See also http://math.ucr.edu/home/baez/week141.html
  * For a nice visualization of the Hopf maps, see https://nilesjohnson.net/hopf.html
 
<tabbox Researcher>

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

--> Example1#

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--> Example2:#

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</tabbox>

Please solve the following equation to prove you're human. 44 +15 = Please keep this field empty:

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