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 2n+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 dimX=2n+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.In this 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 dimX⩽2n+k is estimated and determined for k=0,1,2.