Gottlieb groups of spheres

This paper takes up the systematic study of the Gottlieb groups $G_{n+k}(\S^n)$ of spheres for $k\le 13$ by means of the classical homotopy theory methods. The groups $G_{n+k}(\S^n)$ for $k\le 7$ and $k=10,12,13$ are fully determined. Partial results on $G_{n+k}(\S^n)$ for $k=8,9,11$ are presented as well. We also show that $[\iota_n,\eta^2_n\sigma_{n+2}]=0$ if $n=2^i-7$ for $i\ge 4$.