Lifting to mod 2 Moore spaces

Abstract

We try to investigate whether an element of order $2$ in homotopy groups of spheres has a lift to an element of homotopy groups of mod $2$ Moore spaces or not. A typical element of the affirmative property is Thomeier's element of ${\pi }_{8n+7}\left(S^{4n+3}\right)$. And the Whitehead square $[l_{2n+1},l_{2n+1}]$ of the identity class $l_{2n+1}$ of $S^{2n+1}$ is a negative example except for $n=0,1$ and $3$.