For $n\geq 2$ let $(S^n,p)$ be the $n$-sphere with a base point $p$. Let $1:S^n\to S^n$ denote the identity map. Let us define the map

$Wh_1: \pi_i(S^n,p)\to \pi_{i+n-1}(S^n,p), \alpha\mapsto [\alpha,1],$

where $[\cdot,\cdot]$ is the Whitehead bracket.

**Question 1** : What is known about $Wh_1$ ?

**Question 2** : If we let $Wh_f$ denote the corresponding map on homotopy groups for $f:(S^m,p)\to (S^n,p)$ then what is known about $Wh_f$ ?