{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/on-the-geometric-fixed-points-of-the-real","title":"On the geometric fixed points of the real topological cyclic homology of $\\mathbb{Z}/4$","arxiv_id":"2411.11993","date":"2024-11-18","proceeding":null,"authors":["Thomas Read"],"abstract":"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 $\\pi_i(\\mathrm{TCR}(\\mathbb{Z}/4)^{\\phi \\mathbb{Z}/2})$ 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.","url_abs":"https://arxiv.org/abs/2411.11993v1","url_pdf":"https://arxiv.org/pdf/2411.11993v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"on-the-geometric-fixed-points-of-the-real","repo_url":"https://github.com/thjread/tcr_geo_fixed","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}