Let $a$ in $\mathbb S(\mathbb C^{n})$, the unit sphere in $\mathbb C^n$. Does there exists a continuous map $x\mapsto u_x$, from $\mathbb S(\mathbb C^{n})$ to $U(n)$, the group of unitary endomorphisms of $\mathbb C^{n}$, such that $u_x(a)$ equals $x$ for all $x$?
As far as I can tell (that is not very far), the fact that the composition of $x\mapsto u_x$ and $v\in U(n) \mapsto v(a)$ is the identity on $\mathbb S(\mathbb C^{n})$ does not seem to lead to an obvious contraction about the cohomology spaces. But still, I am unable to define such a map.
And what if we consider the analogous question but with $\mathbb P^{n-1} \to PU(n)$ rather than $\mathbb S(\mathbb C^{n}) \to U(n)$?