Hopf Bundle

Intuitive

Concrete

"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." http://nilesjohnson.net/hopf.html

• The best explanation and especially how it is related to 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

Examples

$S_1 \to S_0$

Real Numbers $\mathbb{R}$ are used to define the Hopf map $S^1 \to S^0$. The fibre $S^0$ here is the group of unit real numbers $\{-1,1\}$, also known as $Z/2$.

$S_3 \to S_2$

Complex Numbers $\mathbb{C}$ are used to define the Hopf map $S^3 \to S^2$. This Hopf map describes a magnetic monopole of unit strength.

$S_7 \to S_4$

Quaternions $\mathbb{H}$ are used to define the Hopf map $S^7 \to S^4$. This Hopf map describes a single instanton.

$S_{15} \to S_8$

Octonions $\mathbb{O}$ are used to define the Hopf map $S^{15} \to S^8$. Currently there is no physics application known of this map! This map is different from the other two, because the fibre $S^7$ (the unit octonions) is not really a group. The reason for this is that octonions aren't associative.

Sources for these examples: Section 9.4 "Principal Bundles" in Geometry, Topology and Physics by Nakahara and http://math.ucr.edu/home/baez/week141.html

(The classification of the Hopf bundles as listed here is surprisingly similar to the classification of all simple Lie groups. Each Hurwitz algbra corresponds to one family of simple groups. The octonions play a special role, because they correspond to the exceptional family, which has only a finite number of members.)

Abstract

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.

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.

https://twitter.com/math3ma/status/837075372617302016

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).

http://math.ucr.edu/home/baez/week141.html

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.

page 16 in Topology, Geometry and Gauge fields by Naber

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

https://ncatlab.org/nlab/show/Hopf+fibration