Products of Greek letter elements dug up from the third Morava stabilizer algebra

TL;DR

This paper investigates the cohomology of the third Morava stabilizer algebra to identify nontrivial products of Greek letter elements in the stable homotopy groups of spheres, extending previous work on the second algebra.

Contribution

It introduces new methods using the third Morava stabilizer algebra's cohomology to find nontrivial products of Greek letter elements in stable homotopy groups.

Findings

Identifies nontrivial products of Greek letter elements in stable homotopy groups.

Extends previous results from the second to the third Morava stabilizer algebra.

Provides explicit conditions for nontriviality involving prime numbers.

Abstract

Oka and the second author considered the cohomology of the second Morava stabilizer algebra to study nontriviality of the products of beta elements of the stable homotopy groups of spheres. In this paper, we use the cohomology of the third Morava stabilizer algebra to find nontrivial products of Greek letters of the stable homotopy groups of spheres: $\al_1\ga_t$, $\be_2\ga_t$, $\lrk{\al_1,\al_1,\be_{p/p}^p}\ga_t\be_1$ and $\lrk{\be_1,p,\ga_t}$ for $t$ with $p\nmid t(t^2-1)$ for a prime number $p>5$.