How to write down a loop on $\mathrm{Diff}(\mathbb{S}^2)$ concretely? If we regard $\mathbb{S^2}$ as the complex projective line $\mathbb{P}^1$, and define a loop on its diffeomorphism group:
$$\gamma: \mathbb{S}^1\longrightarrow\mathrm{Diff}(\mathbb{S}^2)$$
$$z\mapsto([x,y]\mapsto[\bar{z}^kx,y])$$
Where $|z|=1$, $[x,y]$ is the homogeneous coordinate, $k$ is an integer, how to show that this mapping defines a nontrivial loop on $\mathrm{Diff}(\mathbb{S}^2
)$ for $k$ odd?
Here the "nontrivial" means it is not homotopic to the constant loop $\beta(z)=\mathrm{id}_{\mathbb{S}^2}$, i.e. a non trivial element in the fundamental group $\pi_1(\mathrm{Diff}(\mathbb{S}^2
))$.
 A: Thanks for @Jason DeVito's hint, the fundamental group $\pi_1(\mathrm{Diff}(S^2))$ is isomorphic to $\pi_1(\mathrm{SO}(3))$, so now, it suffices to transform this loop into a loop $r$ in $\mathrm{SO}(3)$.
Note that one can identify $\mathbb{P}^1$ with $S^2$ via the map
$$[x,y]\mapsto\left(\frac{\Re(x\bar{y})}{|x|^2+|y^2|},\frac{\Im(x\bar{y})}{|x|^2+|y|^2},\frac{|x|^2-|y|^2}{|x|^2+|y|^2}\right)$$
where the later we use the Euclidean coordinate of the $S^2$, it is not hard to see our loop defined here $\gamma(z)=([x,y]\mapsto[\bar{z}^kx,y])$ corresponds to the coordinate transformation (if we denote $z=e^{i\theta}$):
$$\left(\frac{\Re(x\bar{y})}{|x|^2+|y^2|},\frac{\Im(x\bar{y})}{|x|^2+|y|^2},\frac{|x|^2-|y|^2}{|x|^2+|y|^2}\right)\mapsto \left(\frac{\cos k\theta\cdot\Re(x\bar{y})+\sin k\theta\cdot\Im(x\bar{y})}{|x|^2+|y^2|},\frac{\cos k\theta\cdot\Im(x\bar{y})-\sin k\theta\cdot\Re(x\bar{y})}{|x|^2+|y|^2},\frac{|x|^2-|y|^2}{|x|^2+|y|^2}\right)$$
which write in the transformation matrix is
$$r(\theta)=\begin{pmatrix}\cos k\theta& \sin k\theta& 0\\-\sin k\theta & \cos k\theta & 0\\0&0&1\end{pmatrix}\in\pi_1(\mathrm{SO}(3))$$
We wanna to analyse the triviality of this loop $r$ in $\mathrm{SO}(3)$, we can lift it to the level of $\mathrm{SU}(2)$, one can choose a double cover from $\phi:\mathrm{SU}(2)\longrightarrow\mathrm{SO}(3)$, defined by:
$$\phi(A)=(R(A)_{ij})_{3\times3}$$
where the entry $R(A)_{ij}$ is defined by
$$R(A)_{ij}=\frac{1}{2}\mathrm{tr}(\sigma_iA\sigma_j A^*)$$
where $\sigma_k$ are the Pauli matrices:
$$\sigma_1=\begin{pmatrix}0&1\\1&0\end{pmatrix},\sigma_2=\begin{pmatrix}0&-\sqrt{-1}\\\sqrt{-1}&0\end{pmatrix},\sigma_3=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$$
hence a lifted curve $\tilde{r}$ of $r$ under the covering $\phi$ in $\mathrm{SU}(2)$ is:
$$\tilde{r}(\theta)=\begin{pmatrix}e^{\frac{ik\theta}{2}}&0\\0&e^{-\frac{ik\theta}{2}}\end{pmatrix}$$
It is a contractible curve in $\mathrm{SU}(2)$ if and only if $k$ is even (this can be obtained by putting this curve into $S^3$), hence our $\gamma$ defined on $\mathrm{Diff}(S^2)$ is trivial iff $k$ is even.
