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

TL;DR

This paper explores the cohomology of the Morava stabilizer algebra S(3) and demonstrates the nontriviality of certain products in the stable homotopy groups of spheres, linking Cohen's elements with third periodic elements.

Contribution

It provides new insights into the structure of stable homotopy groups by analyzing the cohomology of S(3) and establishing nontrivial products involving Cohen's elements and third periodic elements.

Findings

Nontrivial products in π_* of spheres for p ≥ 7

Connection between Cohen's elements and third periodic elements

Cohomology results for Morava stabilizer algebra S(3)

Abstract

In this paper, we study the cohomology of the Morava stabilizer algebra $S(3)$. 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 \cite{Co}, $\gamma_s$ is a third periodic homotopy elements.