Determination of the order of the P–image by Toda brackets

Abstract

The present paper gives a proof of the author's paper [14] on the orders of Whitehead products of ι n with α ∈ π n n+k , (n ≥ k + 2, k ≤ 24) and improves and extends it. The method is to use composition methods in the homotopy groups of spheres and rotation groups. 55M35, 55Q52; 57S17

Relations in the 24-th homotopy groups of spheres

“…The main purpose of this note is to give a proof of the fact that the Toda brackets ν, σ, ν and ν, η, σ are not trivial. This result gives an affirmative answer to [12,Conjecture 4.8]. In the proof of Theorem 1.1, our method is to inspect relations in homotopy groups of spheres through those in homotopy groups of rotation groups.…”

“…Theorem 12.16,12.17], we have ζ, η, ν = 0. Hence we obtain the relation(2.32) µ 7 σ 16 ∈ {ζ 7 , η 18 , ν 19 }.…”

“…12 = {ν 12 , σ 20 , ν27 } = {ν 12 , η 15 , σ16 } = σ′ 12 .Proof. 6 + P (ξ 13 )η 29 mod Eπ 5 29 = {δ 6 , μ6 σ 23 , ν 6 σ 9 κ 16 }.Let SO(n) be the n-th rotation group, i k,n : SO…”