On the commutativity of the localized self homotopy groups of [formula omitted

Abstract

For a connected Lie group G, the homotopy set [ G , G ] inherits the group structure by the pointwise multiplication and is called by the self homotopy group of G. In this paper we work with the case G = SU ( n ) , U ( n ) . It was shown by McGibbon that SU ( n ) and U ( n ) themselves are homotopy commutative when they are localized at p and p > 2 n − 1 . Thus the p-localized self homotopy groups of SU ( n ) and U ( n ) are commutative, if p > 2 n − 1 . Then the converse is true? In this paper, we completely determine, for which p, the p-localized self homotopy group of G is commutative, in the case G = U ( n ) , SU ( n ) or SU ( n ) / H where H is a subgroup of the center of SU ( n ) .