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

Published:2004-03-01
Printed: Mar 2004
• Stephen D. Theriault
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## Abstract

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 $G_2$.
 Keywords: Lie group, exponent
 MSC Classifications: 55Q52 - Homotopy groups of special spaces