Revisiting the $\beta_1$-action on the $3$-primary stable homotopy groups of spheres

TL;DR

This paper provides a simplified proof and generalizations of known results about the action of the class _1 on the 3-primary stable homotopy groups of spheres, using advanced spectral sequence tools.

Contribution

It offers a straightforward proof and extends the understanding of the _1 action on 3-primary stable homotopy groups, including generalizations to other 144-periodic families.

Findings

Confirmed _1^5 eq 0 and _1^6 = 0 in stable homotopy groups.

Proved that products of five _{1+9s} are non-zero.

Established that products of six _{1+9s} vanish.

Abstract

Let $\beta_1$ be the first $3$-torsion class in the stable homotopy groups of spheres in even degree. Toda showed that $\beta_1^5 \neq 0$, whilst $\beta_1^6 = 0$. Shimomura generalised this to the $144$-periodic family generated by $\beta_1$, written as $\{\beta_{1+9s}\}_{s\geq 0}$, and showed that any $5$-fold product $\prod_5 \beta_{1+9s} \neq 0$, whilst all $6$-fold products $\prod_6 \beta_{1+9s} = 0$. In this article, we give a simple proof of these results as well as some generalisations to other $144$-periodic families. Our tools include BP-synthetic spectra, and the well-known Adams--Novikov spectral sequence for the spectrum of topological modular forms at the prime $3$ as well as its Adams operations.