Published online by Cambridge University Press: 20 November 2018
Exponent information is proven about the Lie groups $SU(3),\,SU(4),\,Sp(2)$, and
${{G}_{2}}$ by showing some power of the
$H$-space squaring map (on a suitably looped connected-cover) is null homotopic. The upper bounds obtained are
$8,\,32,\,64$, and
${{2}^{8}}$ respectively. This null homotopy is best possible for
$SU(3)$ given the number of loops, off by at most one power of 2 for
$SU(4)$ and
$Sp(2)$, and off by at most two powers of 2 for
${{G}_{2}}$.