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:**


1. If $M = G$ is a Lie group, then a section $s$ exists: for $x\in G$ put $s(x) := \text{left multiplication by }x\cdot x\_{0}^{-1}$.
2. 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 $n=1,3,7$? 



Existence of sections of the evaluation map for the diffeomorphism group
If a section $\sigma : M \rightarrow \text{Diff}\_{+}(M)$ to $\text{Diff}\_{+}(M) \rightarrow M$ exists, then $M$ must be parallelizable (i.e. the tangent bundle of $M$ must be trivial). Indeed, if $\vec{e}\_1,\ldots,\vec{e}\_n$ is a basis for the tangent space of $M$ at $x\_0$, then $\sigma(x)\_{\ast}(e\_1),\ldots,\sigma(x)\_{\ast}(e\_n)$ is a basis for the tangent bundle to $M$ at $x \in M$. These vary continuously with $x$, and thus define a trivialization of the tangent bundle of $M$.


For spheres, a theorem that was proved independently in


Kervaire, Michel A.
Non-parallelizability of the n-sphere for n>7. 
Proc. Natl. Acad. Sci. USA 44 (1958), no. 3, 280–283.


and


Bott, R.; Milnor, J.
On the parallelizability of the spheres. 
Bull. Amer. Math. Soc. 64 1958 87–89. 


says that $S^k$ is only parallelizable when $k=1,3,7$, so your examples are the only ones.


