{"paper":{"title":"No metrics with Positive Scalar Curvatures on Aspherical 5-Manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.DG","authors_text":"Misha Gromov","submitted_at":"2020-09-11T10:48:40Z","abstract_excerpt":"A metric space $X$ is called uniformly acyclic if there there exists an {\\it acyclicty control function} $R=R(r)=R_X(r)\\geq r $, $0\\leq r <\\infty$, such that the homology inclusion homomorphisms between the balls around all points $x\\in X$, $$H_i(B_x(r))\\to H_i(B_x(R))$$  vanish for all $i=1,2,\\ldots$. We show that if a complete orientable $m$-dimensional manifold $\\tilde X$ of dimension $m\\leq 5$ admits a proper (infinity goes to infinity) distance decreasing map to a complete $m$-dimensional uniformly acyclic manifold, then the scalar curvature of $\\tilde X$ can't be uniformly positive, $$\\i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2009.05332","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2009.05332/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}