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 E_2-term of the Adams spectral sequence, Ext_A^{\ast,\ast} (Z_p,Z_p). In this paper we prove that in the cohomology of A, the product k_0 h_n \tilde \delta _{s + 4} \in Ext_A^{s + 7, t(s,n) + s} (Z_p, Z_p), is nontrivial for n \geq 5, and trivial for n=3, 4, where \tilde\delta_{s + 4} is actually \tilde\alpha_{s + 4}^{(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].

Keywords: Steenrod algebra, cohomology, Adams spectral sequence, May spectral sequence

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