Gottlieb groups of spheres

This paper takes up the systematic study of the Gottlieb groups $G_{n+k}\left(S^n\right)$ of spheres for $k\le 13$ by means of the classical homotopy theory methods. We fully determine the groups $G_{n+k}\left(S^n\right)$ for $k\le 13$ except for the 2-primary components in the cases: $k=9,n=53;k=11,n=115$. In particular, we show $[{\iota }_n,{\eta }_n^2{\sigma }_{n+2}]=0$ if $n=2^i-7$ for $i\ge 4$.