$2$-Primary Exponent Bounds for Lie Groups of Low Rank

http://dx.doi.org/10.4153/CMB-2004-013-x
Canad. Math. Bull. 47(2004), 119-132

  Published:2004-03-01
 Printed: Mar 2004

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 $G_2$.