The beta elements \(\beta_{tp^2/r}\) in the homotopy of spheres
The author studies the Adams-Novikov spectral sequence to detect beta elements of the homotopy groups of spheres. Let \(M\) be the mod \(p\) Moore spectrum, and \(V_{a}\) the cofiber of \(\Sigma^{2a(p-1)} M\rightarrow M\), and \(W\) the cofiber of \(\beta_1: S^{2p(p-1)}\to S^0\). The author [Proc. Am. Math. Soc. 138, No.~4, 1495--1499 (2010; Zbl 1207.55013)] proved that if the element \(v_{2}^{s} \in E^{0,2s(p+1)(p-1)}(W\wedge V_{a})\) survives to \(\pi_{2s(p+1)(p-1)}(W\wedge V_{a})\), then \(\overline{\beta}_{st/r}\) survives to \(\pi_*(S^0)\) for \(t>0\) and \( r\leq a-2\). In this paper, the author proves that \(v^{p^{2}}_{2} \in E_{2}^{0, 2p^{2}(p+1)(p-1)}(W \wedge V_{p^{2}})\) survives for \(p\geq 7\). Hence for \(p\geq 7\), beta elements \(\overline{\beta}_{tp^2/r}\) survive to the stable homotopy groups for \(t\geq 1\) and \( r \leq p^2-2\) .
scientific article; zbMATH DE number 5809275