Let $M$ be a closed connected oriented smooth manifold and $\mathrm{Diff}_{+}(M)$ the group of orientation preserving diffeomorphisms of $M$ endowed with the compact-open topology. Pick a base point $x_{0} \in M$ and consider the evaluation map $$\mathrm{ev}\colon \mathrm{Diff}_{+}(M) \to M, \quad \mathrm{ev}(g) = g(x_{0}).$$

I would like to know when the evaluation map admits a global section, i.e. a *continuous* map $s\colon M \to \mathrm{Diff}_{+}(M)$ such that $\mathrm{ev} \circ s = \mathrm{id}_{M}$.

**Examples:**

- If $M = G$ is a Lie group, then a section $s$ exists: for $x\in G$ put $s(x) := \text{left multiplication by }x\cdot x_{0}^{-1}$.
- However, for $M = S^{2}$, the 2-sphere, such a section $s$ cannot exist: it would induce an
*injective*map on homotopy groups $s_{*}\colon \pi_{k}(S^{2}) \to \pi_{k}(\mathrm{Diff}_{+}(S^{2}))$, but $\pi_{2}(S^2) \cong \mathbb{Z}$ while $$\pi_{2}(\mathrm{Diff}_{+}(S^{2})) \cong \pi_{2}(\mathrm{SO}(3)) = 0.$$

It is probably hard to decide on the existence of a section for general $M$, so let us restrict to spheres $S^{n}$:

**Question:** For which $n\in \mathbb{N}$ does the evaluation map $\mathrm{ev}\colon \mathrm{Diff}_{+}(S^{n}) \to S^{n}$ admit a continuous global section?

Since $S^1$ and $S^3$ are naturally Lie groups, the section exists in thsese two cases by the first example above. We can analogously use octonionic multiplication to construct a global section for $S^{7}$. Are there any other cases than $n=1,3,7$?