Symmetric A actions on $\mathcal{A}(2)$
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914929485807616 |
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| author | Bruner, Robert R. |
| author_facet | Bruner, Robert R. |
| contents | We describe the variety of `symmetric' left actions of the mod 2 Steenrod algebra $\mathcal{A}$ on its subalgebra $\mathcal{A}(2)$. These arise as the cohomology of $\text{v}_2$ self maps $Σ^7 Z \longrightarrow Z$, as in arXiv:1608.06250 [math.AT]. There are $256$ $\mathbb{F}_2$ points in this variety, arising from $16$ such actions of $Sq^8$ and, for each such, $16$ actions of $Sq^{16}$. We describe in similar fashion the 1600 $\mathcal{A}$ actions on $\mathcal{A}(2)$ found by Roth(1977) and the inclusion of the variety of symmetric actions into the variety of all actions. We also describe two related varieties of $\mathcal{A}$ actions, the maps between these and the behavior of Spanier-Whitehead duality on these varieties. Finally, we note that the actions which have been used in the literature correspond to the simplest choices, in which all the coordinates equal zero. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_16980 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Symmetric A actions on $\mathcal{A}(2)$ Bruner, Robert R. Algebraic Topology 55S10 (Primary), 55P25, 55-08 We describe the variety of `symmetric' left actions of the mod 2 Steenrod algebra $\mathcal{A}$ on its subalgebra $\mathcal{A}(2)$. These arise as the cohomology of $\text{v}_2$ self maps $Σ^7 Z \longrightarrow Z$, as in arXiv:1608.06250 [math.AT]. There are $256$ $\mathbb{F}_2$ points in this variety, arising from $16$ such actions of $Sq^8$ and, for each such, $16$ actions of $Sq^{16}$. We describe in similar fashion the 1600 $\mathcal{A}$ actions on $\mathcal{A}(2)$ found by Roth(1977) and the inclusion of the variety of symmetric actions into the variety of all actions. We also describe two related varieties of $\mathcal{A}$ actions, the maps between these and the behavior of Spanier-Whitehead duality on these varieties. Finally, we note that the actions which have been used in the literature correspond to the simplest choices, in which all the coordinates equal zero. |
| title | Symmetric A actions on $\mathcal{A}(2)$ |
| topic | Algebraic Topology 55S10 (Primary), 55P25, 55-08 |
| url | https://arxiv.org/abs/2408.16980 |