On the homotopy groups of the suspended quaternionic projective plane and applications

Yang, Juxin; Mukai, Juno; Wu, Jie

doi:10.2140/agt.2025.25.2981

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On the homotopy groups of the suspended quaternionic projective plane and applications

Juxin Yang, Juno Mukai and Jie Wu

Algebraic & Geometric Topology 25 (2025) 2981–3033

DOI: 10.2140/agt.2025.25.2981

Abstract

We determine the $(2, 3)$ -primary components of the homotopy groups $π_{r + k} (Σ^{k} ℍ P^{2})$ for all $7 \leq r \leq 15$ and $k \geq 0$ . Essentially, we give the determinations of the integral homotopy groups $π_{r + k} (Σ^{k} ℍ P^{2})$ for all $0 \leq r \leq 15$ and $k \geq 0$ , which in particular include the unstable ones. As applications, we first construct a suspended generalized Hopf fibration by a Toda bracket localized at $2$ . Then we provide two classification theorems for CW complexes of type $S^{4 + k} \cup e^{8 + k} \cup e^{12 + k}$ (for $k \geq 1$ but $k \neq 4$ ) localized at 3, one of which states that the homotopy type of the space ${(Σ^{k} ℍ P^{3})}_{(3)}$ as a CW complex of the above type depends only on its Steenrod module. Our results also yield some wedge decompositions of $3$ -local suspended self smash products of quaternionic projective spaces.

Keywords

homotopy group, Gray's relative James construction, Toda bracket, quaternionic projective plane

Mathematical Subject Classification

Primary: 55Q52

References

Publication

Received: 13 November 2023

Revised: 17 May 2024

Accepted: 16 June 2024

Published: 17 September 2025

Authors

Juxin Yang
School of Mathematical Sciences Hebei Normal University Shijiazhuang China	Beijing Key Laboratory of Topological Statistics and Applications for Complex Systems Beijing Institute of Mathematical Sciences and Applications Beijing China

Juno Mukai
Shinshu University Matsumoto Japan

Jie Wu
Beijing Institute of Mathematical Sciences and Applications Beijing China

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Volume 25, issue 5 (2025)