{"paper":{"title":"Relative group cohomology and the orbit category","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.AT","authors_text":"Ergun Yalcin, Semra Pamuk","submitted_at":"2012-05-05T11:26:25Z","abstract_excerpt":"Let $G$ be a finite group and $\\cF$ be a family of subgroups of $G$ closed under conjugation and taking subgroups. We consider the question whether there exists a periodic relative $\\cF$-projective resolution for $\\bbZ$ when $\\cF$ is the family of all subgroups $H \\leq G$ with $\\rk H \\leq \\rk G-1$. We answer this question negatively by calculating the relative group cohomology $\\cF H^* (G, \\bbF_2)$ where $G=\\bbZ/2\\times \\bbZ /2$ and $\\cF$ is the family of cyclic subgroups of $G$. To do this calculation we first observe that the relative group cohomology $\\cF H^*(G, M)$ can be calculated using "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.1120","kind":"arxiv","version":1},"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"}