A nontrivial product of filtration \(s+ 5\) in the stable homotopy of spheres
Let \(\beta_{s+2}\) and \(\gamma_{p^{n-2}/p^{n-2}-1}\) be the beta and the gamma elements of the stable homotopy groups of spheres at the prime \(p\) greater than three constructed by \textit{L. Smith} [Am. J. Math. 99, 418-436 (1977; Zbl 0389.55017)] and \textit{J. Lin} [Hiroshima Math. J. 31, No.3, 477-492 (2001; Zbl 0997.55018)], respectively. In this paper, the author essentially shows the non-triviality of the product of these elements by observing the \(E_2\)-term of the Adams spectral sequence. The main tool he uses to determine the \(E_2\)-term is the May spectral sequence. First he shows that the element \(a_2^sh_{2,0}h_{1,1}\) of the \(E_2\)-term detects the beta element, and then the non-triviality of the product \(a_2^sh_{2,0}h_{1}(b_0h_n +h_1b_{n-1})\) in the \(E_2\)-term. Here \(a_2^s\) denotes the \(s\)-fold Yoneda product of \(a_2\), and the element \(b_0h_n +h_1b_{n-1}\) in the \(E_2\)-term detects the gamma element, \textit{J. Lin} [op. cit.]. The main result, which says that \(\beta_{s+2}\gamma_{p^{n-2}/p^{n-2}-1}\) is essential in the stable homotopy of spheres for \(n\geq 3\), \(0\leq s< p-3\), follows by showing that the original \(E_2\)-terms of the Adams differentials that may hit the product are all trivial.
scientific article; zbMATH DE number 5162316