For some reason, today I want to understand better the group of diffeomorphisms of the 2-sphere, $S^2$.

After a few minutes I found this result by Smale in 1958.

The space $\Omega$ of all orientation-preserving $C^\infty$ diffeomorphisms of $S^2$ has a strong deformation retract [to] the rotation group $SO(3)$.

It sounds like any invertible, differentiable map $f:S^2 \to S^2$ can be "smoothed" to a 3D rotation of a certain angle $\theta$ around a certain axis $\vec{k}$.

Smale gives an explicit retraction from $f \in C^\infty(S^2)$ to the rotation

$$\begin{array}{rcc} f(x_0) &\mapsto& x_0 \\ df(e_1) &\mapsto& e_1 \\ df(e_2) &\mapsto& e_2 \end{array}$$

There seems to be an alternative proof in some topology notes. Jacob Lurie describes a fibration over the space of all conformal structures on the 2-sphere

\[ \mathrm{Diff}_{\mathrm{Conf}}(S^2) \to \mathrm{Diff}(S^2) \to \mathrm{Conf}(S^2) \]

and another diagram related to the existence of isothermal coordinates:

\[ \begin{array}{ccccc} SO(2) & \to & SO(3) & \to & S^2 \\ \downarrow & & \downarrow & & \downarrow \\ Aut(\mathbb{C}) & \to & \mathrm{Diff}_{\mathrm{Conf}}(S^2) & \to & S^2 \end{array} \]

These discussions of the diffeomorphism group 2-sphere were not very concrete, but after writing this I have answered much of my own questions...

I suppose can could get conformal structures like $e^{f(z)} dz\, d\overline{z}$.

What kind of information did we lose after these reductions?

What are some elements of $\mathrm{Diff}(S^2)$ you can build?

**REMARK**:
One comment suggested I use vector fields to construct my diffeomorphism. If I have a smooth section of the tangent bundle $T(S^2)$ can it be "integrated" to a diffeomorphism of $S^2 \to S^2$ ?