Revisiting the β₁-action on the 3-primary stable homotopy groups of spheres

2025/11/04 by Davies, Jack Morgan
#55N20 #55N34 #55P42 #55Q10 #55Q51 #55T05 #55T25 #Algebraic Topology (math.AT) #FOS: Mathematics #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.2511.02324

Abstract

Let β₁ be the first 3-torsion class in the stable homotopy groups of spheres in even degree. Toda showed that β₁⁵ ≠ 0, whilst β₁⁶ = 0. Shimomura generalised this to the 144-periodic family generated by β₁, written as \β_1+9s\_{s≥ 0}, and showed that any 5-fold product ∏₅ β_1+9s ≠ 0, whilst all 6-fold products ∏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.

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Related

• Nonvanishing of products in v₂-periodic families at the prime 3
• Stable homotopy groups of spheres: From dimension 0 to 90
• Classical stable homotopy groups of spheres via \mathbbF₂-synthetic methods
• On periodic families in the stable stems of height two
• The triviality of the 61-stem in the stable homotopy groups of spheres