Nilpotency of unstable K-theory

In the preceding papers [H. Hamanaka, A. Kono, On [ X , U ( n ) ] , when dimX is 2n, J. Math. Kyoto Univ. 43 (2) (2003) 333–348; H. Hamanaka, On [ X , U ( n ) ] , when dimX is 2 n + 1 , J. Math. Kyoto Univ. 44 (3) (2004) 655–667; H. Hamanaka, Adams e-invariant, Toda bracket and [ X , U ( n ) ] , J. Math. Kyoto Univ. 43 (4) (2003) 815–828], the group structure of the homotopy set [ X , U ( n ) ] with the pointwise multiplication is studied, where X is a finite CW-complex and U ( n ) is the unitary group. It is seen that nil [ X , U ( n ) ] = 2 for some X with its dimension 2n, and, when dim X = 2 n + 1 and n is even, [ X , U ( n ) ] is expressed as the two stage central extension of an Abelian group, i.e., nil [ X , U ( n ) ] ⩽ 3 . s paper, we consider the nilpotency class of [ X , U ( n ) ] , especially, for given k, the maximum of the nil [ X , U ( n ) ] under the condition dim X ⩽ 2 n + k is estimated and determined for k = 0 , 1 , 2 .