{"paper":{"title":"The loop cohomology of a space with the polynomial cohomology algebra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Samson Saneblidze","submitted_at":"2008-10-24T19:22:49Z","abstract_excerpt":"Given a simply connected space $X$ with the cohomology $H^*(X;{\\mathbb Z}_2)$ to be polynomial, we calculate the loop cohomology algebra $H^*(\\Omega X;{\\mathbb Z}_2)$ by means of the action of the Steenrod cohomology operation $Sq_1$ on $H^*(X;{\\mathbb Z}_2).$ As a consequence we obtain that $H^*(\\Omega X;{\\mathbb Z}_2)$ is the exterior algebra if and only if $Sq_1$ is multiplicatively decomposable on $H^{\\ast}(X;{\\mathbb Z}_2).$ The last statement in fact contains a converse of a theorem of A. Borel."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0810.4531","kind":"arxiv","version":5},"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"}