Relations in the 24-th homotopy groups of spheres

TL;DR

This paper proves certain non-trivial Toda brackets in the 24th homotopy groups of spheres and establishes a specific relation among elements in these groups, advancing understanding of their algebraic structure.

Contribution

It confirms a conjecture about the non-triviality of specific Toda brackets and clarifies relations among elements in the 24th homotopy groups of spheres.

Findings

Proves the non-triviality of Toda brackets <ν̄,σ,ν̄> and <ν,η,σ̄>.

Establishes the relation ν̄₇ω₁₅=ν₇σ₁₀κ₁₇ in π₃₁⁷.

Supports the conjecture on the determination of the P-image by Toda brackets.

Abstract

The main purpose of this note is to give a proof of the fact that the Toda brackets $<\bar{\nu},\sigma,\bar{\nu}>$ and $<\nu,\eta, \bar{\sigma}>$ are not trivial. This is an affirmative answer of the second author's Conjecture (Determination of the $P$-image by Toda brackets, Geometry and Topology Monographs 13(2008), 355-383). The second purpose is to show the relation $\bar{\nu}_7\omega_{15}=\nu_7\sigma_{10}\kappa_{17}$ in $\pi^7_{31}$.