3. CR maps

Problem 3.1.
[J. Dâ€™Angelo] Let $f\colon S^{2n1}\to S^{2N1}$ and $g\colon S^{2n1}\to S^{2N1}$ be two CR maps with $N$ minimal; i.e. $f$ and $g$ are not CR maps into a lower dimensional sphere. How can one show that $f$ and $g$ are not homotopic in target dimension $2N1$?
Remark. There are four CR maps from $S^3$ to $S^5$ which are not homotopic as maps to $S^5$:
\begin{align*} (z,w) & \mapsto (z,w,0), \\ (z,w) & \mapsto (z^2,w^2,\sqrt{2}zw), \\ (z,w) & \mapsto (z^2,zw,w), \\ (z,w) & \mapsto (z^3,\sqrt{3}zw, w^3) . \end{align*}
However, these maps are all homotopic as maps into $S^7$. 
Remark. [J. Dâ€™Angelo] A smooth CR map in this context must be rational.

Cite this as: AimPL: Analysis and geometry on pseudohermitian manifolds, available at http://aimpl.org/pshermitian.