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

TL;DR

This paper investigates the homotopy types and automorphism groups of the suspension of the infinite quaternionic projective space, establishing uniqueness of its n-type and analyzing the finiteness and structure of its automorphism groups.

Contribution

It proves the uniqueness of the homotopy type with a given n-type and characterizes the automorphism groups as finite or infinite depending on n, revealing non-abelian structures.

Findings

Homotopy type with the same n-type is unique.

Automorphism group is finite for n ≤ 9.

Automorphism group is infinite and non-abelian for n ≥ 13.

Abstract

Let $[\rho_{i_k},[\rho_{i_{k-1}},...,[\rho_{i_{1}}, \rho_{i_2}] ...]]$ be an iterated commutator of self-maps $\rho_{i_j} : \Sigma {\Bbb H}P^\infty \to \Sigma {\Bbb H}P^\infty, j = 1,2, ..., k$ 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 $\text{Aut}(\pi_{\leq n} (\Sigma {\Bbb H}P^\infty)/\text{torsion})$ of automorphisms is finite for $n \leq 9$, and infinite for $n \geq 13$, and that $\text{Aut}(\pi_{*} (\Sigma {\Bbb H}P^\infty)/\text{torsion})$ becomes non-abelian.