Some products involving the fourth Greek letter family element $tilde{delta_s}$ in the Adams spectral sequence

Abstract

Let p be an odd prime and A be the mod p Steenrod algebra. For computing the stable homotopy groups of spheres with the classical Adams spectral sequence, we must compute the E2-term of the Adams spectral sequence, $Ext^{∗,∗}_A(Bbb{Z}_p,Bbb{Z}_p)$ . In this paper we prove that in the cohomology of A, the product $k_0h_ntilde{delta}_{s+4} in Ext_A^{s+7,t(s,n}+s}(Bbb{Z}_p,Bbb{Z}_p)$, is nontrivial for $n geq 5$, and trivial for n =3,4, where $tilde{delta}_{s+4}$ is actually $tilde{alpha}^{(4)}_{s+4}$ described by Wang and Zheng, $p geq 11, 0 leq s < p−4$ and $t(s, n)=2(p−1)[(s+2)+(s+4)p+(s+3)p^2+(s+4)p^3+p^n]$.