A relation in the stable homotopy groups of spheres

Abstract

Let p ⩾ 7 be an odd prime. Based on the Toda bracket 〈α ₁ β ₁ ^p−1, α ₁ β ₁, p,γ _s〉, the authors show that the relation α ₁ β ₁ ^p−1 h _2,0 γ _s = β _p/p−1 γ _s holds. As a result, they can obtain α ₁ β ₁ ^p h _2,0 γ _s = 0 ∈ π _*(S ⁰) for 2 ⩽ s ⩽ p − 2, even though α ₁ h _2,0 γ _s and β ₁ α ₁ h _2,0 γ _s are not trivial. They also prove that β ₁ ^p−1 α ₁ h _2,0 γ ₃ is nontrivial in π _*(S ⁰) and conjecture that β ₁ ^p−1 α ₁ h _2,0 γ _s is nontrivial in π _*(S ⁰) for 3 ⩽ s ⩽ p − 2. Moreover, it is known that \beta _{p/p - 1} \gamma _3 = 0 \in Ext_{BP_* BP}^{5,*} (BP_* ,BP_* ), but β _p/p−1 γ ₃ is nontrivial in π _*(S ⁰) and represents the element β ₁ ^p−1 α ₁ h _2,0 γ ₃.

Keywords

Toda bracket / Stable homotopy groups of spheres / Adams-Novikov spectral sequence / Method of infinite descent

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