{"paper":{"title":"$\\mathrm{G}$-theory of $\\mathbb{F}_1$-algebras I: the equivariant Nishida problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.KT","math.RA"],"primary_cat":"math.AT","authors_text":"Snigdhayan Mahanta","submitted_at":"2011-10-27T07:54:47Z","abstract_excerpt":"We develop a version of $\\mathrm{G}$-theory for an $\\mathbb{F}_1$-algebra (i.e., the $\\mathrm{K}$-theory of pointed $G$-sets for a pointed monoid $G$) and establish its first properties. We construct a Cartan assembly map to compare the Chu--Morava $\\mathrm{K}$-theory for finite pointed groups with our $\\mathrm{G}$-theory. We compute the $\\mathrm{G}$-theory groups for finite pointed groups in terms of stable homotopy of some classifying spaces. We introduce certain Loday--Whitehead groups over $\\mathbb{F}_1$ that admit functorial maps into classical Whitehead groups under some reasonable hypot"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.6001","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}