Detecting nontrivial products in the stable homotopy ring of spheres via the third Morava stabilizer algebra (Q6602177)

In the stable homotopy ring \(\pi_*(S)\) of the \(p\)-local sphere spectrum \(S\) for a prime \(p\ge 7\), there are the Greek letter elements \(\alpha_s\), \(\beta_s\) and \(\gamma_s\) and the Cohen's elements \(\zeta_s\) for \(s\ge 1\). The authors consider the non-triviality of products of these generators. For this sake, they study the degree preserving ring homomorphism \(\phi\colon E_2^{*}\to H^*S(3)\). Here, \(E_2^*\) denotes the \(E_2\)-page of the Adams-Novikov spectral sequence \(E_2^*=Ext^*_{BP_*BP}(BP_*,BP_*)\Rightarrow \pi_*(S)\) based on the Brown-Peterson spectrum \(BP\), \(S(3)\) is the third Morava stabilizer algebra, and the structure of \(H^*S(3)\) is completely determined. Their argument is as follows: For \(\xi_i\in\{\alpha_s, \beta_s, \gamma_s, \zeta_s:s\ge 1\}\), take an element \(\overline \xi_i\in E_2^*\) that detects it. If \(\phi(\prod_i \overline\xi_i)\ne 0\), then not only \(\prod_i\overline \xi_i\ne 0\), but nothing hits it under any differential of the Adams-Novikov spectral sequence. Indeed, such an element has the Adams filtration less than 10. It follows that \(\prod_i\xi_i\ne 0\in \pi_*(S)\). Under this framework, they find all such non-trivial products. Among them, the new results are \( \beta_1\gamma_s\zeta_n\ne 0\in \pi_*(S) \) for \(n\equiv 2\) mod 3 and \(s\not\equiv 0,\pm 1\) mod \(p\). They also correct some typos in the previous papers on \(H^*S(3)\).