Hopf fiber

Basically, the Hopf fiber is a fiber structure on the 2-sphere S^2. Or, it describes a decomposition of S^3 into S^2 and S^1.

Start from the definition of S^3\subset \mathbb{C}^2, it consists of points (z,z')\in\mathbb{C}^2 such that |z|^2+|z'|^2=1. On defines a projection p:S^3\rightarrow S^2,(z,z')\mapsto (2\bar{z}z',|z|^2-|z'|^2) where 2\bar{z}z' stands for the first coordinates of points in S^2. Note that, the image of p indeed lies in S^2. We will show that p is surjective and for each P\in S^2, the fiber p^{-1}(P) is a circle. And this describes a fiber structure on S^2.

Indeed, for any (x+iy,w)\in S^2, we want to find (z,z') such that

|z|^2+|z'|^2=1,|z|^2-|z'|^2=w,2\bar{z}z'=x+iy

Yet this is almost trivial, since we have that

|z|^2=(1+w)/2,|z'|^2=(1-w)/2,2\bar{z}z'=x+iy

This system is always solvable. Moreover, if p(z,z')=(x+iy,w), then p^{-1}(x+iy,w)=(e^{i\theta}z,e^{i\theta}z'). This can also be seen from the above system of equations. Note that sometimes we are likely to take this projection, p':S^3\rightarrow S^2,(z,z')\mapsto (z,|z'|). One problem with this projection is that, its image is not the entire S^2(|z'| is nover negative), another point is that p'^{-1}(p'(z,z'))=p'^{-1}(z,|z'|)=(z,e^{i\alpha}z'), yet for the point (z,0)\in S^2, its fiber is only itself, only one point, not a circle.

Another interesting way to describe this fiber is to use some Lie groups. Note that, for S^3, the Lie group SU(2) acts transitively and freely on S^3(because each matrix in SU(2) can be written as \begin{pmatrix}a &b \\ -\bar{b} & \bar{a} \end{pmatrix}, in general, a matrix in U(2) can be written as e^{i\theta}\begin{pmatrix} a &b\\-\bar{b}&\bar{a}\end{pmatrix}).

But, SU(2) acts also mysteriously on \mathbb{R}^3! This is a rather important fact. Note that, in some sense, this is also reasonable: that is SU(2) acts also on the tangent space of some point P\in S^3 on S^3, and S^3 has a group structure(that is exactly SU(2)). After giving this tangent space T_P(S^3) a metric(induced from that of S^3, of course), SU(2) acts thus on T_P(S^3)=\mathbb{R}^3 preserving the metric, thus acts also on S^2\subset \mathbb{R}^3. Yet, we know that, SO(3) acts on S^2, we can show that the action of SU(2) on S^2 induces a homomorphism SU(2)\rightarrow SO(3), with kernel \pm Id. Moreover, SO(3) acts transitively on S^2, and for each point P\in S^2, the stabilizer of SO(3) on P is a circle. Note also that, for the opposite point -P\in S^2, the stabilizer of SO(3) on -P is the same as the previous, Stab(SO(3),P)=Stab(SO(3),-P). This shows that, not rigorously, that SU(2) acts on S^2 with for each point, a stabilizer group the same as one copy of circle.

More concretely, since SU(2) consists of matrices of the form \begin{pmatrix}a&b\\-\bar{b}&\bar{a}\end{pmatrix} with |a|^2+|b|^2=1, we consider the conjugate action of SU(2) on itself. This is indeed a group action, and we note this as

Ad_g:SU(2)\rightarrow SU(2),h\mapsto ghg^{-1}

Now take derivative with respect to the variable h, and transform this action to the Lie algebra

ad_g:\mathfrak{g}(SU(2))\rightarrow \mathfrak{g}(SU(2)),H\mapsto gHg^{-1}

Note that, the Lie algebra of SU(2) consists of matrices H\in M_{2\times 2}(\mathbb{C}) such that Tr(H)=0,\bar{H}^t+H=0. More specifically, we can find a \mathbb{R}-basis for \mathfrak{su}(2)=\mathfrak{g}(SU(2))(note that, SU(2) is not a complex Lie group, only a real Lie group, thus its Lie algebra is only a real vector space, not a complex one):

e_1=\begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix},    e_2=\begin{pmatrix}0&i\\i&0\end{pmatrix},e_3=\begin{pmatrix}0&1\\-1&0\end{pmatrix}

We can show that the metric induced from SU(2)=S^3 on \mathfrak{su}(2) is preserved by SU(2), and e_1,e_2,e_3 forms an orthonormal basis. Now, we want to show that the stabilizer of each point P\in S^2=S^2(\mathfrak{su}(2)) is a circle, since SU(2) acts transitively on S^2, we can verify this for just one point, say e_3. Note that, if g\in SU(2) preserves e_3, then g=r_{\theta}=\begin{pmatrix} \cos{\theta} & \sin{\theta} \\ -\sin{\theta} &\cos{\theta}\end{pmatrix}(\theta\in\mathbb{R}). So, after some routine yet simple calculations, we find that, under the basis e_1,e_2, ad_{r_{\theta}}=\begin{pmatrix} \cos{2\theta} & \sin{2\theta} \\ -\sin{2\theta} &\cos{2\theta}\end{pmatrix}(why, you may wonder, there is an increase in the ‘speed’? This can be explained by taking another derivative with respect to g in ad_g, thus we get that ad_G(H)=[G,H]=GH-HG. For g=r_{\theta}, we have that G=\frac{dr_{\theta}}{d\theta}|_{\theta=0}=\begin{pmatrix}0&1\\-1&0\end{pmatrix}=e_3. It is readily verified that ad_{e_3}(e_1)=-2e_2,ad_{e_3}(e_2)=2e_1, this little 2 is the origin for this speed increase! And this 2 accounts also for the fact that \pi_1(S^3)=\pi_1(SU(2))=0,\pi_1(SO(3))=\pi_1(\mathbb{P}^3_{\mathbb{R}})=\mathbb{Z}/2).

So, in this way, we describe another way of the Hopf fiber on S^2.

The second method enables us to generalize to higher dimension case.

A nice demonstration of the Hopf fiber is here.

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