Hopf Fibrations-Construction and Quaternion Interpretations (I)

Before I begin discussing the Hopf fibration of the 3-spbhere, one of the simplest yet deeply profound example of a non-trivial fiber bundle, I’d like to recall the definition of a fiber bundle.

Let E,B,F represent the entire space, base space and the fiber respectively where E,B are connected. If f:E \mapsto B is a continuous surjection onto the base space, then the structure (E,B,F,f) is said to be a fiber bundle if for every x\in E, there exists a neighborhood U \subset B of f(x) such that there exists a homeomorphism \psi:f^{-1}(U) \mapsto U \times F such that \psi \circ \pi_{U}=f.

What this basically means is that locally, the fiber bundle looks like the product B \times F but globally, it may have different topological properties.

A trivial fiber bundle is a fiber bundle which in which the total space is E=B \times F. In fact, any fiber bundle over a contractible CW Complex is trivial.

Hopf Fibration

The 3-sphere is the set of points (x_{1},x_{2},x_{3},x_{4}) in \mathbb{R}^{4} which satisfies x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=1.

It was proven by Heinz Hopf that the 3-sphere an be constructed as a non-trivial fiber bundle.

Here, the base space is S_{2} and the fiber space is the circle,S_{1}.

So, the fiber bundle structure is:

hopf

What’s really amazing is that this fiber bundle is very non-trivial. Watch this video on youtube to see a stereographic projection of S_{3} on \mathbb{R}^{3}. Every circle in the projection is linked to every other circle in the manner of a Villarceau circle obtained by cutting a torus through its center at a particular angle. The stereographic projection preserves circles except the circle at the projection point , which is mapped to a ‘line of infinity’ or more appropriately, a circle of infinity.

hopf_fibration

The tori are formed because a circle in S_{2} has the fiber circle S_{1} linked to each point.

We know that \mathbb{R}^{4} \cong \mathbb{C}^{2} and \mathbb{R}^{3} \cong \mathbb{C} \times \mathbb{R}. A typical point of \mathbb{C}^{2} is (z_{1},z_{2}) and for \mathbb{C} \times \mathbb{R} is (z,x).

The original projection devised by Hopf was the following:

\pi(z_{1},z_{2})=(2z_{1} \bar{z_{2}},|z_{1}|^{2}-|z_{2}|^{2}) where |z_{1}|^{2}+|z_{2}|^{2}=1.

Clearly, the second component is real whereas the first is complex when z_{1} \neq z_{2}. Since this is a projection on S_{2}, it is easy to see that

|2z_{1}\bar{z_{2}}|^{2}+(|z_{1}|^{2}-|z_{2}|^{2})^{2}=1.       (1)

Now, that the projection on S_{2} has been settled, there remains to determine the fiber(which we already know to be S_{1}).

So, let \pi(z_{1},z_{2})=\pi(w_{1},w_{2} for some distinct points (z_{1},z_{2}),(w_{1},w_{2}) \in S_{3}.

From (1):

\Rightarrow (w_{1},w_{2})= (\lambda z_{1},\lambda z_{2}) for some complex number \lambda with |\lambda|=1. Also, every point (\lambda z_{1},\lambda z_{2}) where (z_{1},z_{2}) \in S_{3} and |\lambda|=1.

Obviously the locus of points \lambda \in \mathbb{C} which satisfies |\lambda|=1 is a circle. Hence, the fiber F=S_{1} is a circle.

Quaternionic Interpretation

A quaternion q is often represented as:

q=a+b \textbf{i}+c\textbf{j}+d\textbf{k}

where \textbf{i}, \textbf{j}, \textbf{k} are the fundamental quaternion units. The essence of most of the arguments that follow will be the fact that the utility of quaternions in spatial rotations can be exploited to study the fiber bundle structure of S_{3}.

A versor is merely a quaternion with unit norm.

An  imaginary quaternion is one in which the real part(i.e a in the above expression) is zero.

A vector in \mathbb{R}^{3} can be identified as an imaginary quaternion.

The group of versors/unit quaternions ic represented as Sp(1) and is isomorphic to SU(2).

Given any versor/unit quaternion, it is well known that p \mapsto qp \bar{q} is a rotation of the vector p \in \mathbb{R}^{3} (see this for more information). So now, we study the action of the group of versors on the vectors in \mathbb{R}.

Fix an imaginary quaternion p=k_{1} \textbf{i}+k_{2} \textbf{j}+k_{3} \textbf{k}. Clearly, the action:

q \mapsto qp\bar{q}

defines a surjective homomorphism from S_{3} to S_{2} if p is a point on S_{2}. Now the main question is:which versor fixes the arbitrary vector p?

Consider two real numbers x,y which satisfy x^{2}+y^{2}=1. The versor q=x+yp fixes p under the above map. Now obviously, this stabilizer of p is isomorphic to S_{1}.

Other fibrations involving only spheres?

Now that we have determined the fiber bundle structure of S_{3} where the total space, base space and the fiber are all n-spheres, the question arises if there are any other similar examples.

An obvious one is S_{1} which has a trivial surjective homomorphism onto itself and the fiber is S_{0} which is just a point.

Well, there actually is another example and that is known as the quaternionic Hopf fibration. It is a fibration of S_{7} over S_{4} where the fiber is S_{3}.

hopf_2

Notice that S_{7} can be thought of as a sphere on \mathbb{H} \times \mathbb{H}, so we can define a surjective homomorphism pi: S_{7} \mapsto \mathbb{HP}^{1} by \pi(x,y)=[x;y] where x,y \in \mathbb{H} and |x|^{2}+|y|^{2}=1.

Multiplication on \mathbb{H}^{2} \mapsto \mathbb{H} can be defined on the sphere in \mathbb{H} in an obvious manner. So, factoring out by S_{3}, one gets \mathbb{HP}^{1}, the quaternionic projective line. Now, the main observation is that S_{4} \cong \mathbb{HP}^{1} which completes the discussion on the fiber structure.

So, now are there any other examples. Yes, another example is the fiber bundle of S_{15} over S_{8} where the fiber is S_{7}. Viewing S_{8} as the octonionic projective line, \mathbb{OP}^{1}, the fiber bundle structure in an analogous manner. This is called the Octonionic Hopf fibration.

Can we exploit similarities and hope for a generalization or at least a few more examples?

Sadly, the answer is no. In fact,  a theorem of Frank Adam proves that these are the only fiber bundles where all the spaces involved are spheres through Hopf invariants. With the help of machinery from co-homology theory and the existenc eof the four normed division algebras, we’ll be to unravel Adam’s theorem. This is just the first part of a series of posts which will deal with Hopf fibrations, constructions, invariants and related concepts.

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