What can be said about rational maps $S^2\to S^2$, $S^3\to S^3$, etc, which are invertible and have rational inverses? (Here, $S^n$ denotes the subset $\{x\in\mathbb{R}^{n+1}:|x|=1\}$.)
I know that $\pi_n(S^n)$ is $\mathbb{Z}$, and I suspect that this implies that the only injective maps $S^n\to S^n$ are homotopic to the identity function.
I have also run across the notion of Cremona groups, but have not been able to fruitfully use this information. For example, the map $[x:y:z]\mapsto[y^2+z^2:xy:xz]$ is a beautiful involution of $P(\mathbb{R}^2)$, but I don't know if this can be modified to be a map $S^2\to S^2$. (In particular, I would like a version of the map which extends to a map of the balls bounded by $S^2$.)
Any reference which considers this question or states that this question is open or hard would be greatly appreciated.