On automorphisms of graded quasi-Lie algebras

Abstract

Let $\mathbb Z$ be the ring of integers and let $K({\Bbb Z}, 2n)$ denote the Eilenberg-MacLane space of type $({\Bbb Z}, 2n)$ for $n \geq 1$.

In this article, we prove that the graded group

$$A_m: = \text{Aut}(\pi_{\leq 2mn+1} (\Sigma K({\Bbb Z}, 2n))/\text{torsions})$$ of automorphisms of the graded quasi-Lie algebras $\pi_{\leq 2mn+1} (\Sigma K({\Bbb Z}, 2n))$ modulo torsions that preserve the Whitehead products is a finite group for $m \leq 2$ and an infinite group for $m \geq 3$,

and that the group $\text{Aut} (\pi_{*} (\Sigma K({\Bbb Z}, 2n))/\text{torsions})$ is non-abelian. We extend and apply those results to techniques in localization (or rationalization) theory.