On Samelson products in p-localized unitary groups

Abstract

The unitary group U ( n ) has elements ε i ∈ π 2 i + 1 ( U ( n ) ) ( 0 ⩽ i ⩽ n − 1 ) of its homotopy groups in the stable range. In this paper we show that certain multi Samelson products of type 〈 ε i , 〈 ε j , ε k 〉 〉 are non-trivial. This leads us to the result that the nilpotency class of the group of the self homotopy set [ SU ( n ) , SU ( n ) ] is no less than 3, if 4 ⩽ n . Also by the power of generalized Samelson products, we can see the further result that, for a prime p and an integer n = p k , nil [ SU ( n ) , SU ( n ) ] ( p ) ⩾ 3 , if (1) p ⩾ 7 or (2) p = 5 and n ≡ 0 or 1 mod 4 .