The composition of R. Cohen's elements and the third periodic elements in stable homotopy groups of spheres

Abstract

In this paper, we re-compute the cohomology of the Morava stabilizer algebra $S(3)$ [12, 16]. As an application, we show that for $p \geq 7$, if $s\not \equiv 0, \pm 1 \,\, mod \,(p) $, $n\not \equiv 1 \,\, mod\, 3$, $n>1$, then $\zeta_n\gamma_s$ is a nontrivial product in $\pi_*(S)$ by Adams-Novikov spectral sequence, where $\zeta_n$ is created by R. Cohen [1], $\gamma_s$ is a third periodic homotopy elements.