On automorphisms of graded quasi-lie algebras

Let Z be the ring of integers and let K(Z, 2n) denote the Eilenberg-MacLane space of type (Z, 2n) for n ≥ 1. In this article, we prove that the graded group A_m:= Aut(π_≤2mn+1 (ΣK(Z, 2n))/torsions) of automorphisms of the graded quasi-Lie algebras π_≤2mn+1 (ΣK(Z, 2n)) modulo torsions that preserve the Whitehead products is a finite group for m ≤ 2 and an infinite group for m ≥ 3, and that the group Aut(π_∗(ΣK(Z, 2n))/torsions) is non-abelian. We extend and apply those results to techniques in localization (or rationalization) theory.