A note on the Greek letter elements of the homotopy of the $L_n$-local spheres
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908431558901760 |
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| author | Kato, Ryo Shimomura, Katsumi Shimomura, Mao-no-suke |
| author_facet | Kato, Ryo Shimomura, Katsumi Shimomura, Mao-no-suke |
| contents | Let $BP$ denote the Brown-Peterson spectrum at a prime $p$, whose homotopy groups are isomorphic to the polynomial algebra generated by elements $v_i$'s for $i\ge 1$. We consider the homotopy groups of the $v_n^{-1}BP$-localized sphere spectrum $L_nS^0$ with $n^2\le 2p-1$, and show that the groups contain the $n$-th Greek letter family. For the proof of this, we further show the existence of the $v_n^{-1}BP$-localized Smith-Toda spectrum $W_n$ for the case $n^2\le 2p-1$. If the Smith-Toda spectrum $V(n-1)$ exists, then $L_nV(n-1)$ is an example of $W_n$. Previously, $W_n$ is shown to exist if $n^2+n\le 2p-1$.
We also consider the case where $(p,n)=(7,4)$, and show the existence of the delta family in $π_*(L_4S^0)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_02155 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on the Greek letter elements of the homotopy of the $L_n$-local spheres Kato, Ryo Shimomura, Katsumi Shimomura, Mao-no-suke Algebraic Topology 55Q45 (Primary) 55Q51 (Secondary) Let $BP$ denote the Brown-Peterson spectrum at a prime $p$, whose homotopy groups are isomorphic to the polynomial algebra generated by elements $v_i$'s for $i\ge 1$. We consider the homotopy groups of the $v_n^{-1}BP$-localized sphere spectrum $L_nS^0$ with $n^2\le 2p-1$, and show that the groups contain the $n$-th Greek letter family. For the proof of this, we further show the existence of the $v_n^{-1}BP$-localized Smith-Toda spectrum $W_n$ for the case $n^2\le 2p-1$. If the Smith-Toda spectrum $V(n-1)$ exists, then $L_nV(n-1)$ is an example of $W_n$. Previously, $W_n$ is shown to exist if $n^2+n\le 2p-1$. We also consider the case where $(p,n)=(7,4)$, and show the existence of the delta family in $π_*(L_4S^0)$. |
| title | A note on the Greek letter elements of the homotopy of the $L_n$-local spheres |
| topic | Algebraic Topology 55Q45 (Primary) 55Q51 (Secondary) |
| url | https://arxiv.org/abs/2507.02155 |