{"paper":{"title":"Cohomological Dimension, Connectivity, and Lusternik--Schnirelmann category","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Yuli Rudyak","submitted_at":"2017-03-10T18:37:55Z","abstract_excerpt":"Dranishnikov~\\cite{D2} proved that \\[{\\rm cat} X\\leq {\\rm cd}(\\pi_1(X))+\\Bigl\\lceil\\frac{{\\rm hd} (X)-1}{2}\\Bigr\\rceil.\\] where ${\\rm cd}(\\pi)$ denotes the cohomological dimension of a group $\\pi$ and ${\\rm hd}(X)$ denotes the homotopy dimension of $X$. Furthermore, there is a well-known inequality of Grossman,~\\cite{G}: \\[ {\\rm cat} X\\leq \\Bigl\\lceil\\frac{{\\rm hd} (X)}{k+1}\\Bigr\\rceil \\text{ if } \\pi_i(X)=0 \\text{ for } i\\leq k. \\] We make a synthesis and generalization of both of these results, by demonstrating the main result: \\[ {\\rm cat}\\leq {\\rm cd}(\\pi_1(X))+\\Bigl\\lceil\\frac{{\\rm hd} (X"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.03788","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"}