Some products involving the fourth Greek letter family element δ̃ s in the Adams spectral sequence ∗

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 ( p, p) . In this paper we prove that in the cohomology of A , the product k0hnδ̃s+4 ∈ Ext s+7,t(s,n)+s A ( p, p) , is nontrivial for n ≥ 5, and trivial for n = 3, 4, where δ̃s+4 is actually α̃ (4) s+4 described by Wang and Zheng, p ≥ 11, 0 ≤ s < p−4 and t(s, n) = 2(p−1)[(s+2)+(s+4)p+(s+3)p+(s+4)p+p] . Key word and phrases: Steenrod algebra, cohomology, Adams spectral sequence, May spectral sequence.