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

Abstract

In this paper, we study the cohomology of the Morava stabilizer algebra $S(3)$. As an application, we show that for $p\ge 7$, if $s\not\equiv 0,\pm 1modp$, $n\not\equiv 1mod3$, $n>1$, then ${\zeta }_n{\gamma }_s$ is a nontrivial product in ${\pi }_{\ast }(S)$ by Adams-Novikov spectral sequence, where ${\zeta }_n$ is created by R. Cohen \cite{Co}, ${\gamma }_s$ is a third periodic homotopy elements.

16pages, small corrections, to appear in Osaka Journal of Mathematics