advanced_tools:hopf_bundle
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advanced_tools:hopf_bundle [2017/12/20 10:45] jakobadmin [Examples] |
advanced_tools:hopf_bundle [2018/05/03 11:07] jakobadmin [Concrete] |
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| ====== Hopf Bundle ====== | ====== Hopf Bundle ====== | ||
| - | <tabbox Why is it interesting?> | + | <tabbox Intuitive> |
| - | + | ||
| - | 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). | + | |
| - | + | ||
| - | <cite>http://math.ucr.edu/home/baez/week141.html</cite> | + | |
| - | </blockquote> | + | |
| - | + | ||
| - | <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> | + | |
| - | + | ||
| - | + | ||
| - | + | ||
| - | <tabbox Layman> | + | |
| <note tip> | <note tip> | ||
| Line 37: | Line 7: | ||
| </note> | </note> | ||
| | | ||
| - | <tabbox Student> | + | <tabbox Concrete> |
| Line 50: | Line 20: | ||
| * See also http://math.ucr.edu/home/baez/week141.html | * See also http://math.ucr.edu/home/baez/week141.html | ||
| * For a nice visualization of the Hopf maps, see https://nilesjohnson.net/hopf.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> | ||
| - | | + | ---- |
| - | <tabbox Examples> | + | |
| + | **Examples** | ||
| + | |||
| --> $S_1 \to S_0$# | --> $S_1 \to S_0$# | ||
| Line 97: | Line 65: | ||
| - | <tabbox FAQ> | + | <tabbox Abstract> |
| - | + | ||
| - | <tabbox History> | + | <note tip> |
| + | The motto in this section is: //the higher the level of abstraction, the better//. | ||
| + | </note> | ||
| + | |||
| + | <tabbox Why is it interesting?> | ||
| + | |||
| + | 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). | ||
| + | |||
| + | <cite>http://math.ucr.edu/home/baez/week141.html</cite> | ||
| + | </blockquote> | ||
| + | |||
| + | <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> | ||
| </tabbox> | </tabbox> | ||
advanced_tools/hopf_bundle.txt · Last modified: 2018/05/03 11:07 by jakobadmin
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