nLab real Hopf fibration

Idea

The real Hopf fibration is the fibration

$$\array{ S^0 &\hookrightarrow& S^1 \\ && \downarrow^{\mathrlap{p_{\mathbb{R}}}} \\ && S^1 }$$

of the 1-sphere over itself with fiber the 0-sphere, which is induced via the Hopf construction from the product operation

$$\mathbb{R} \times \mathbb{R} \stackrel{(-)\cdot (-)^{-1}}{\longrightarrow} \mathbb{R}$$

on the real numbers.

This may also be understood as the Spin(2)-double cover of SO(2).

Realizations

Here are different but equivalent ways of realizing this explicitly:

Via projective space

If the domain $S^1$ is regarded as the unit sphere $\{(x,y) | {\vert x\vert}^2 + {\vert y\vert}^2 = 1\}$ in $\mathbb{R}\times \mathbb{R}$ and the codomain $S^1$ is regarded as the real projective space, then $p$ is given simply by

$$p \colon (x,y) \mapsto [x;y] = [x/y; 1] \,,$$

One can view the real Hopf fibration as the boundary of a Möbius strip, which is the non-trivial double cover of the circle.

As an element in the first (stable) homotopy group of spheres $\pi_1(S^1) \simeq \mathbb{Z}$, the real Hopf fibration represents $p_{\mathbb{R}} = 2$.

Via join construction

We spell out the real Hopf fibration realized as the Hopf construction

$$\array{ X \ast X &\overset{H_{\mathbb{R}}}{\longrightarrow}& \Sigma X \\ = && = \\ (X \times [0,1] \times X)_{/\sim} && (X \times [0,1])_{/\sim} \\ (x,t,y) &\mapsto& \big( f(x,y), t\big) }$$

(as defined there) on

$$X \coloneqq \mathbb{Z}/2$$

regarded with its cyclic group-structure

$$\array{ (\mathbb{Z}/2) \times (\mathbb{Z}/2) &\overset{f}{\longrightarrow}& (\mathbb{Z}/2) \\ (x,y) &\mapsto& x + y }$$

Here the equivalence relation for the suspension $\Sigma X = (X \times [0,1])_{/\sim}$ on the right is

$$(x_1,0) \sim (x_2,0) \;\,,\;\; (x_1, 1) \sim (x_2, 1) \;\;\; \forall x_1,x_2 \in X$$

while the equivalence relation for the join $X\ast X = (X \times [0,1] \times X)_{/\sim}$ on the left is

$$(x, 0, y_1) \simeq (x,0,y_2) \;\;,\;\; (x_1,1,y) \sim (x_2, 1, y) \,.$$

So on the left we have a circle $S^1$ realized by gluing four copies of the interval $[0,1]$ labeled in $\mathbb{Z}/2 \times \mathbb{Z}/2$ by identifying them pairwise at $0 \in [0,1]$ and pairwise the other way at $1 \in [0,1]$. while on the right we have a circle $S^1$ realized by two copies, labeled by $\mathbb{Z}/2$.

$$S^1 \longrightarrow S^1 \,.$$

A full path around the circle on the left is given, in terms of the above coordinates in $[0,1] \times (\mathbb{Z}_2 \times \mathbb{Z}_2)$, by

$$\array{ (0,(0,0)) &\rightsquigarrow& (1,(0,0)) \\ && \sim \\ && (1,(1,0)) &\rightsquigarrow& (0,(1,0)) \\ &&&& \sim \\ &&&& (0,(1,1)) &\rightsquigarrow& (1,(1,1)) \\ &&&&&& \sim \\ &&&&&& (1,(0,1)) & \rightsquigarrow& (0,(0,1)) \\ &&&&&&&& \sim \\ &&&&&&&& (0,(0,0)) }$$

As we map this path over to the other $S^1$ by adding up the two coordinate labels in $\mathbb{Z}/2$, we trace out the following path on the right, with coordinates in $[0,1] \times \mathbb{Z}/2$:

$$\array{ (0,(0+0 = 0)) &\rightsquigarrow& (1,(0 + 0 = 0)) \\ && \sim \\ && (1,(1 + 0 = 1)) &\rightsquigarrow& (0,(1 + 0 = 1)) \\ && && \sim \\ && && (0,(1+1 = 0)) &\rightsquigarrow& (1,(1+1 = 0)) \\ && && && \sim \\ && && && (1,(0 + 1 = 1)) &\rightsquigarrow& (0,(0 + 1 = 1)) \\ && && && && \sim \\ && && && && (0,(0 + 0 = 0)) }$$

That’s twice around the circle on the right, for once on the left, manifestly showing that the real Hopf fibration is the non-trivial double cover of the circle by itself:

$$S^1 \overset{\cdot 2}{\longrightarrow} S^1 \,.$$

Related concepts

• complex Hopf fibration

• quaternionic Hopf fibration

• octonionic Hopf fibration