On the same $N$-type of the suspension of the infinite quaternionic projective space

Abstract

Let $[{\rho }_{i_k},[{\rho }_{i_{k-1}},...,[{\rho }_{i_1},{\rho }_{i_2}]...]]$ be an iterated commutator of self-maps ${\rho }_{i_j}:\Sigma \mathbb{H}{P}^{\infty }\to \Sigma \mathbb{H}{P}^{\infty },j=1,2,...,k$ on the suspension of the infinite quaternionic projective space. In this paper, it is shown that the image of the homomorphism induced by the adjoint of this commutator is both primitive and decomposable. The main result in this paper asserts that the set of all homotopy types of spaces having the same $n$-type as the suspension of the infinite quaternionic projective space is the one element set consisting of a single homotopy type. Moreover, it is also shown that the group $\text{Aut}({\pi }_{\le n}(\Sigma \mathbb{H}{P}^{\infty })/\text{torsion})$ of automorphisms is finite for $n\le 9$, and infinite for $n\ge 13$, and that $\text{Aut}({\pi }_{\ast }(\Sigma \mathbb{H}{P}^{\infty })/\text{torsion})$ becomes non-abelian.