English

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

Algebraic Topology 2007-05-23 v1

Abstract

The complex affine quadric Qm={zCm+1z12+...+zm+12=1}Q^{m}=\{z\in {\Bbb C}^{m+1}\mid z_{1}^{2}+...+z_{m+1}^{2}=1\} deforms by retraction onto SmS^{m}; this allows us to identify [Qk,Qn][Q^{k},Q^{n}] and [Sk,Sn]=πk(Sn)[S^{k},S^{n}]=\pi_{k}(S^{n}). Thus one will say that an element of πk(Sn)\pi_{k}(S^{n}) is complex representable if there exists a complex polynomial map from QkQ^{k} to QnQ^{n} corresponding to this class. In this Note we show that πn+1(Sn)\pi_{n+1}(S^{n}) and πn+2(Sn)\pi_{n+2}(S^{n}) are complex representable.

Keywords

Cite

@article{arxiv.math/0702324,
  title  = {Complex Polynomial Representation of $\pi_{n+1}(s^{n})$ and $\pi_{n+2}(s^{n})$},
  author = {Francisco-Javier Turiel},
  journal= {arXiv preprint arXiv:math/0702324},
  year   = {2007}
}

Comments

5 pages, no figures