Topological Hochschild homology of the image of j
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866911345338744832 |
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| author | Lee, David Jongwon Levy, Ishan |
| author_facet | Lee, David Jongwon Levy, Ishan |
| contents | We compute the mod $(p,v_1)$ and mod $(2,η,v_1)$ $\mathrm{THH}$ of many variants of the image-of-$J$ spectrum. In particular, we do this for $j_ζ$, whose $\mathrm{TC}$ is closely related to the $K$-theory of the $K(1)$-local sphere. We find in particular that the failure for $\mathrm{THH}$ to satisfy $\mathbb{Z}_p$-Galois descent for the extension $j_ζ \to \ell_p$ corresponds to the failure of the $p$-adic circle to be its own free loop space. For $p>2$, we also prove the Segal conjecture for $j_ζ$, and we compute the $K$-theory of the $K(1)$-local sphere in degrees $\leq 4p-6$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_04248 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Topological Hochschild homology of the image of j Lee, David Jongwon Levy, Ishan Algebraic Topology K-Theory and Homology We compute the mod $(p,v_1)$ and mod $(2,η,v_1)$ $\mathrm{THH}$ of many variants of the image-of-$J$ spectrum. In particular, we do this for $j_ζ$, whose $\mathrm{TC}$ is closely related to the $K$-theory of the $K(1)$-local sphere. We find in particular that the failure for $\mathrm{THH}$ to satisfy $\mathbb{Z}_p$-Galois descent for the extension $j_ζ \to \ell_p$ corresponds to the failure of the $p$-adic circle to be its own free loop space. For $p>2$, we also prove the Segal conjecture for $j_ζ$, and we compute the $K$-theory of the $K(1)$-local sphere in degrees $\leq 4p-6$. |
| title | Topological Hochschild homology of the image of j |
| topic | Algebraic Topology K-Theory and Homology |
| url | https://arxiv.org/abs/2307.04248 |