Revisiting the $β_1$-action on the $3$-primary stable homotopy groups of spheres
Jack Morgan Davies
TL;DR
This paper analyzes the action of the first $3$-torsion class $\beta_1$ on the $3$-local stable homotopy groups of spheres, focusing on the height-$2$ divided $β$-family and the $144$-periodic family $\{β_{1+9s}\}$. It provides simple proofs of vanishing and nonvanishing of products in these periodic families using BP-synthetic spectra, the Adams--Novikov spectral sequence for the sphere and for TMF at prime $3$, and a modified ANSS for the equaliser $\mathrm{J}^2$ of Adams operations on TMF. The main results establish upper bounds on nonzero products (e.g., up to five factors for $\prod β_{1+9s}$) and exact vanishing for higher products, while also demonstrating nonvanishing phenomena detected via $\mathrm{J}^2$ and TMF data for a broad class of periodic families. These findings refine our understanding of the multiplicative structure near $β_1$ and provide a robust synthetic framework that can extend to other $144$-periodic families and higher heights, with potential implications for exotic periodicities in synthetic and chromatic contexts.
Abstract
Let $β_1$ be the first $3$-torsion class in the stable homotopy groups of spheres in even degree. Toda showed that $β_1^5 \neq 0$, whilst $β_1^6 = 0$. Shimomura generalised this to the $144$-periodic family generated by $β_1$, written as $\{β_{1+9s}\}_{s\geq 0}$, and showed that any $5$-fold product $\prod_5 β_{1+9s} \neq 0$, whilst all $6$-fold products $\prod_6 β_{1+9s} = 0$. 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.