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

Abstract

The complex affine quadric $Q^m=\{z\in {\mathbb{C}}^{m+1}\mid z_1^2+...+z_{m+1}^2=1\}$ deforms by retraction onto $S^m$; this allows us to identify $[Q^k,Q^n]$ and $[S^k,S^n]={\pi }_k(S^n)$. Thus one will say that an element of ${\pi }_k(S^n)$ 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 ${\pi }_{n+1}(S^n)$ and ${\pi }_{n+2}(S^n)$ are complex representable.

arXiv comment5 pages, no figures