Revisiting the $β_1$-action on the $3$-primary stable homotopy groups of spheres
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917058080407552 |
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| author | Davies, Jack Morgan |
| author_facet | Davies, Jack Morgan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_02324 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Revisiting the $β_1$-action on the $3$-primary stable homotopy groups of spheres Davies, Jack Morgan Algebraic Topology K-Theory and Homology 55T05, 55T25, 55N20, 55N34, 55P42, 55Q10, 55Q51 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. |
| title | Revisiting the $β_1$-action on the $3$-primary stable homotopy groups of spheres |
| topic | Algebraic Topology K-Theory and Homology 55T05, 55T25, 55N20, 55N34, 55P42, 55Q10, 55Q51 |
| url | https://arxiv.org/abs/2511.02324 |