On the degree 2 map for a sphere

The purpose of this article is to compare the two self-maps of $\Omega^kS^{2n+1}$ given by $\Omega^k[2]$ the $k$-fold looping of a degree 2 map and $\Psi^k(2)$ the H-space squaring map. The main results give that in case $2n+1 \neq 2^j-1$, these maps are frequently not homotopic and also that their homotopy theoretic fibres are not homotopy equivalent. The methods are a computation of an unstable secondary operation constructed by Brown and Peterson in the first case and the Nishida relations in the second case. One question left unanswered here is whether the maps $\Omega^{2n+1}[2]$ and $\Psi^{2n+1}(2)$ are homotopic on the level of $\Omega^{2n+1}_0S^{2n+1}$. A natural conjecture is that these two maps are homotopic.