The secondary periodic element \(\beta_{p^{2}/p^{2}-1}\) and its applications

Let p ⩾ 7 be a prime. We prove that \(\beta_{p^{2}/p^{2}-1}\) $${\beta }_{p^2/p^2-1}$$ survives to E_∞ in the Adams-Novikov spectral sequence. Additionally, using the Thom map \(\Phi:\text{Ext}_{BP_{\ast}BP}^{\ast,\ast}(BP_{\ast},BP_{\ast})\rightarrow\text{Ext}_{A}^{\ast,\ast}(\mathbb{Z}/p,\mathbb{Z}/p)\) $$\mathrm{\Phi }:{\text{Ext}}_{B{P}_{\ast }BP}^{\ast ,\ast }(B{P}_{\ast },B{P}_{\ast })\to {\text{Ext}}_A^{\ast ,\ast }(\mathbb{Z}/p,\mathbb{Z}/p)$$ , we can see that h₀h₃ also survives to E_∞ in the classical Adams spectral sequence. As an application of these results, we prove that \(\beta_{p/p}^{p}\) $${\beta }_{p/p}^p$$ is divisible by β₁.