Complex Polynomial Representation of $π_{n+1}(s^{n})$ and $π_{n+2}(s^{n})$

Abstract

The complex affine quadric $Q^{m}=\{z\in {\Bbb C}^{m+1}\mid z_{1}^{2}+...+z_{m+1}^{2}=1\}$ deforms by retraction onto $S^{m}$; this allows us to identify $[Q^{k},Q^{n}]$ and $[S^{k},S^{n}]=π_{k}(S^{n})$. Thus one will say that an element of $π_{k}(S^{n})$ is complex representable if there exists a complex polynomial map from $Q^{k}$ to $Q^{n}$ corresponding to this class. In this Note we show that $π_{n+1}(S^{n})$ and $π_{n+2}(S^{n})$ are complex representable.