A product involving the $\beta$-family in stable homotopy theory

TL;DR

This paper proves the non-triviality of certain composites involving the $eta$-family in the stable homotopy groups of spheres, using combinatorial analysis of spectral sequences.

Contribution

It demonstrates the non-triviality of specific composites involving the $eta$-family in stable homotopy groups, extending previous constructions.

Findings

Non-triviality of $eta$-family composites in stable homotopy groups.

Explicit combinatorial analysis of May spectral sequence.

Extension of known $eta$-family results.

Abstract

In the stable homotopy groups $\pi_{q(p^n+p^m+1)-3}(S)$ of the sphere spectrum $S$ localized at the prime $p$ greater than three, J. Lin constructed an essential family $\xi_{m,n}$ for $n \geq m + 2 >5$. In this paper, the authors show that the composite $\xi_{m,n}\beta_{s}\in \pi_{q(p^n+p^m+sp+s)-5}(S)$ for $2 \leq s < p$ is non-trivial, where $q=2(p-1)$ and $\beta_s \in \pi_{q(sp+s-1)-2}(S)$ is the known $\beta$-family. We show our result by explicit combinatorial analysis of the (modified) May spectral sequence.