Double connectedness of SO3 via action on S2

Question

I understand that $SO(3)$ is doubly-connected and has $SU(2)$ as simply connected double cover.

Now I am trying to understand if I can see this double-connectedness by looking at group actions.

Let us consider the natural action of $SO(3)$ on ${\Bbb S}^2$. Is there a way that I can understand the double-connectedness by looking at the orbits of this action? Or probably at the orbits of an action of a different space than ${\Bbb S}^2$.

I see a certain connection with Group action on simply connected space with simply connected stabilizer but this question has never been answered.

Answer 1

$\DeclareMathOperator{\SO}{SO}$ I will give a proof of $\pi_1(\SO(3))\simeq \mathbb{Z}/2$ using the canonical action of $\SO(3)$ on $S^2$ and a bit of homotopy theory.

The canonical action of $\SO(3)$ on $S^2$ is transitive, so it gives a surjection $$ \pi\colon \SO(3)\to S^2;\quad g\mapsto g \begin{pmatrix} 0\\0\\-1 \end{pmatrix}. $$ The inverse image of $x\in S^2$ is the stabilizer of $x$, which is a coset of $\SO(2)\subset \SO(3)$. Therefore $\pi$ is a fiber bundle with fiber $\SO(2)$. Consider the Serre exact sequence: $$ \pi_2(S^2)\xrightarrow{\delta} \pi_1(\SO(2))\to \pi_1(\SO(3))\to \pi_1(S^2). $$ We use the following facts:

• $\pi_2(S^2)\simeq \mathbb{Z}$ is generated by the class of the map $$ p\colon D^2\to S^2;\quad \begin{pmatrix} r\cos\theta\\r\sin\theta \end{pmatrix}\mapsto \begin{pmatrix} \sin(r\pi)\cos\theta\\ \sin(r\pi)\sin\theta\\ \cos(r\pi) \end{pmatrix}. $$
• $\pi_1(\SO(2))\simeq \mathbb{Z}$ is generated by the class of the map $$ q\colon [0,2\pi]\to \SO(2);\quad \theta\mapsto R_\theta= \begin{pmatrix} \cos \theta&-\sin \theta\\ \sin \theta&\cos \theta \end{pmatrix}. $$
• $\pi_1(S^2)=0$.

It remains to prove $\delta[p]=2[q]$. Consider the map $$ f\colon D^2\to \SO(3);\quad \begin{pmatrix} r\cos\theta\\r\sin\theta \end{pmatrix}\mapsto \begin{pmatrix} R_\theta&0\\ 0&1 \end{pmatrix} \begin{pmatrix} \cos(r\pi)&0&\sin(r\pi)\\ 0&1&0\\ -\sin(r\pi)&0&\cos(r\pi) \end{pmatrix} \begin{pmatrix} R_{-\theta}&0\\ 0&1 \end{pmatrix} \begin{pmatrix} -1&0&0\\ 0&1&0\\ 0&0&-1 \end{pmatrix}. $$ One can easily check that the action of $f(x)\in \SO(3)$ on $S^2$ sends the basepoint $ \begin{pmatrix} 0\\0\\-1 \end{pmatrix}\in S^2 $ to $p(x)$. By the construction of the Serre exact sequence, $\delta[p]$ is given by the class of the map $$ [0,2\pi]\to \SO(2);\quad \theta\mapsto f \begin{pmatrix} \cos \theta\\ \sin \theta \end{pmatrix} =R_{2\theta}, $$ so we get $\delta[p]=2[q]$.