Detection of a nontrivial element in the stable homotopy groups of spheres

Abstract

‎Let $p$ be a prime with $pgeq 7$ and $q=2(p-1)$‎. ‎In this paper‎ <br />‎we prove the existence of a nontrivial product of‎ <br />‎filtration $s+4$ in the stable homotopy groups of spheres‎. ‎This nontrivial‎ <br />‎product is shown to be represented up to a nonzero scalar by‎ <br />‎the product element $widetilde{gamma}_{s}b_{n-1}g_{0}in‎ <br />‎{Ext}_{mathcal{A}}^{s+4,(p^n+sp^2+sp+s)q+s-3}(mathbb{Z}/p,mathbb{Z}/p)$‎ <br />‎in the Adams spectral sequence where $ngeq 2$ and $3leq sleq p-1$‎.