On the geometric fixed points of the real topological cyclic homology of $\mathbb{Z}/4$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910021733842944 |
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| author | Read, Thomas |
| author_facet | Read, Thomas |
| contents | We study the homotopy groups of the geometric fixed points of the real topological cyclic homology of $\mathbb{Z}/4$. We relate these groups to the values of the non-abelian derived functors of the functor $M \mapsto (M \otimes_{\mathbb{Z}/4} M)^{C_2}$ at the $\mathbb{Z}/4$-module $\mathbb{Z}/2$, which we precisely calculate with computer assistance up to degree $6$, and calculate in general up to slight remaining ambiguity. Using these results we compute $π_i(\mathrm{TCR}(\mathbb{Z}/4)^{ϕ\mathbb{Z}/2})$ exactly for $i \le 1$, up to an extension problem for $2 \le i \le 5$, and describe the asymptotic growth of this group for large $i$. A consequence of these computations is that there exists some $0 \le i \le 5$ such that the canonical map comparing the genuine symmetric and symmetric $L$-theory spectra of $\mathbb{Z}/4$ is not an isomorphism on degree $i$ homotopy, and moreover this comparison map is never an isomorphism on homotopy in sufficiently large degrees. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11993 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the geometric fixed points of the real topological cyclic homology of $\mathbb{Z}/4$ Read, Thomas Algebraic Topology K-Theory and Homology 19D55, 11E70 (Primary) 55P91 (Secondary) We study the homotopy groups of the geometric fixed points of the real topological cyclic homology of $\mathbb{Z}/4$. We relate these groups to the values of the non-abelian derived functors of the functor $M \mapsto (M \otimes_{\mathbb{Z}/4} M)^{C_2}$ at the $\mathbb{Z}/4$-module $\mathbb{Z}/2$, which we precisely calculate with computer assistance up to degree $6$, and calculate in general up to slight remaining ambiguity. Using these results we compute $π_i(\mathrm{TCR}(\mathbb{Z}/4)^{ϕ\mathbb{Z}/2})$ exactly for $i \le 1$, up to an extension problem for $2 \le i \le 5$, and describe the asymptotic growth of this group for large $i$. A consequence of these computations is that there exists some $0 \le i \le 5$ such that the canonical map comparing the genuine symmetric and symmetric $L$-theory spectra of $\mathbb{Z}/4$ is not an isomorphism on degree $i$ homotopy, and moreover this comparison map is never an isomorphism on homotopy in sufficiently large degrees. |
| title | On the geometric fixed points of the real topological cyclic homology of $\mathbb{Z}/4$ |
| topic | Algebraic Topology K-Theory and Homology 19D55, 11E70 (Primary) 55P91 (Secondary) |
| url | https://arxiv.org/abs/2411.11993 |