Detecting nontrivial products in the stable homotopy ring of spheres via the third Morava stabilizer algebra

Let p ≥ 7 p \geq 7 be a prime number. Let S ( 3 ) S(3) denote the third Morava stabilizer algebra. In recent years, Kato-Shimomura and Gu-Wang-Wu found several families of nontrivial products in the stable homotopy ring of spheres π ∗ ( S ) \pi _* (S) using H ∗ , ∗ ( S ( 3 ) ) H^{*,*} (S(3)) . In this paper, we determine all nontrivial products in π ∗ ( S ) \pi _* (S) of the Greek letter family elements α s , β s , γ s \alpha _s, \beta _s, \gamma _s and Cohen’s elements ζ n \zeta _n which are detectable by H ∗ , ∗ ( S ( 3 ) ) H^{*,*} (S(3)) . In particular, we show β 1 γ s ζ n ≠ 0 ∈ π ∗ ( S ) \beta _1 \gamma _s \zeta _n \neq 0 \in \pi _*(S) , if n ≡ 2 n \equiv 2 mod 3, s ≢ 0 , ± 1 s \not \equiv 0, \pm 1 mod p p .