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

Abstract

In this paper, we prove that there exists a new family of nontrivial homotopy elements in the stable homotopy groups of spheres with dimension $q\left(p^n+sp+2\right)-4$. These nontrivial homotopy elements are represented by ${\tilde{\beta }}_s{h}_0{h}_n$ in the $E_2^{s+2,t}$–term of the Adams spectral sequence, where $p\ge 5$, $n>4$, $p+1<s<2p-1$, $t=q\left(p^n+sp+2\right)+s-2$, $q=2\left(p-1\right)$.