Relations in the 24-th homotopy groups of spheres

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 M.~Mahowald's Conjecture (J. Mukai, Determination of the $P$-image by Toda brackets, Geometry and Topology Monographs \textbf{13}(2008), 355--383). The second purpose is to determine the relations including ${\bar{\nu }}_6{\omega }_{14}$ in ${\pi }_{30}^6$ and ${\bar{\nu }}_7{\omega }_{15}$ in ${\pi }_{31}^7$. To this end, we provide relations between the Toda bracket and the $J$-homomorphism, and between the Toda bracket and the generalized $P$-homomorphism.

26 pages