The p-primary upsilon1-periodic homotopy groups of a topological space X, denoted by u-11p* Xp , are roughly the parts of the homotopy groups of X localized at a prime p which are detected by K-theory. We will use combinatorial number theory to determine, for p an odd prime, the values of n for which u-11p2 n-1SUn p≅ Z/pn-1+np &parl0;&fll0;np&flr0; !&parr0;. As a corollary, we obtain new bounds for the p-exponent of pi*(SU(n)).
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