
## 3. CR maps

1. #### Problem 3.1.

[J. D’Angelo] Let $f\colon S^{2n-1}\to S^{2N-1}$ and $g\colon S^{2n-1}\to S^{2N-1}$ 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 $2N-1$?
The proposed approach to this problem is to find a homotopy invariant for a family $f_t\colon S^{2n-1}\to S^{2N-1}$. If found, such an invariant could be regarded as a CR analogue of the winding number.
1. 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. D’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.