A new infinite family in the stable homotopy of spheres

In the stable homotopy group π 2 p n + 1 − 2 p n + 2 p 2 − 3 ( V ( 1 ) ) π _{2pⁿ⁺¹-2p^n+2p^2-3}(V(1)) of the Smith-Toda spectrum V ( 1 ) V(1) at a prime p ≥ 5 p≥ 5 , an essential element ϖ n _n was constructed for n ≥ 3 n≥ 3 by X. Liu [Chinese Ann. Math. Ser. B 29 (2008), pp. 291–316]. In this paper, we show that the composite β 1 2 β s − 1 ∗ ϖ n β _1^2β ₛ₋₁ _n is nontrivial, where n > 3 n>3 , 2 ≤ s > p − 2 2≤ s>p-2 , p > 5 p>5 and β s − 1 ∗ β ₛ₋₁ is a β β -related homotopy element in [ V ( 1 ) , S ] 2 ( s − 1 ) p 2 − 2 ( s − 1 ) − 2 p [V (1), S]2(s-1)p^2-2(s-1)-2p . As a corollary, β s − 1 ∗ ϖ n β ₛ₋₁ _n and β 1 β s − 1 ∗ ϖ n β _1β ₛ₋₁ _n are both nontrivial.