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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, $$\\i"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2009.05332","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2020-09-11T10:48:40Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"fda069e8654117bc38f4febafd21ac36aa69ccf9052ce1c35ad4c2ca0455dd60","abstract_canon_sha256":"ddb71b704e8dead9aee75670488d13c6f79fee58943c26cd36b4878f7ded0498"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:34:44.550443Z","signature_b64":"foX/NqWmjH9f+MPsWKYkBDD1R1TE7eh07pa/DajnpWaHpNahm346UxbZ0rJDG6syGF2sljruuMsjmDTiGTxHCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"98a31a59a400a2ef4a400acaf089c0df3e22ba9dd70b4769211ee36c739e307c","last_reissued_at":"2026-07-05T01:34:44.550039Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:34:44.550039Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"No metrics with Positive Scalar Curvatures on Aspherical 5-Manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.DG","authors_text":"Misha Gromov","submitted_at":"2020-09-11T10:48:40Z","abstract_excerpt":"A metric space $X$ is called uniformly acyclic if there there exists an {\\it acyclicty control function} $R=R(r)=R_X(r)\\geq r $, $0\\leq r <\\infty$, such that the homology inclusion homomorphisms between the balls around all points $x\\in X$, $$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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2009.05332","created_at":"2026-07-05T01:34:44.550095+00:00"},{"alias_kind":"arxiv_version","alias_value":"2009.05332v1","created_at":"2026-07-05T01:34:44.550095+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2009.05332","created_at":"2026-07-05T01:34:44.550095+00:00"},{"alias_kind":"pith_short_12","alias_value":"TCRRUWNEACRO","created_at":"2026-07-05T01:34:44.550095+00:00"},{"alias_kind":"pith_short_16","alias_value":"TCRRUWNEACRO6SSA","created_at":"2026-07-05T01:34:44.550095+00:00"},{"alias_kind":"pith_short_8","alias_value":"TCRRUWNE","created_at":"2026-07-05T01:34:44.550095+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":3,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.19619","citing_title":"Some constructions of uniformly positive scalar curvature metrics on open manifolds","ref_index":10,"is_internal_anchor":false},{"citing_arxiv_id":"2606.29135","citing_title":"Gromov's Simplicial Volume Vanishing Conjecture for Positive Scalar Curvature and Fundamental Group Decay","ref_index":21,"is_internal_anchor":false},{"citing_arxiv_id":"2605.20178","citing_title":"Sharp systolic inequalities for K\\\"ahler manifolds","ref_index":122,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/TCRRUWNEACRO6SSABLFPBCOA34","json":"https://pith.science/pith/TCRRUWNEACRO6SSABLFPBCOA34.json","graph_json":"https://pith.science/api/pith-number/TCRRUWNEACRO6SSABLFPBCOA34/graph.json","events_json":"https://pith.science/api/pith-number/TCRRUWNEACRO6SSABLFPBCOA34/events.json","paper":"https://pith.science/paper/TCRRUWNE"},"agent_actions":{"view_html":"https://pith.science/pith/TCRRUWNEACRO6SSABLFPBCOA34","download_json":"https://pith.science/pith/TCRRUWNEACRO6SSABLFPBCOA34.json","view_paper":"https://pith.science/paper/TCRRUWNE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2009.05332&json=true","fetch_graph":"https://pith.science/api/pith-number/TCRRUWNEACRO6SSABLFPBCOA34/graph.json","fetch_events":"https://pith.science/api/pith-number/TCRRUWNEACRO6SSABLFPBCOA34/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TCRRUWNEACRO6SSABLFPBCOA34/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TCRRUWNEACRO6SSABLFPBCOA34/action/storage_attestation","attest_author":"https://pith.science/pith/TCRRUWNEACRO6SSABLFPBCOA34/action/author_attestation","sign_citation":"https://pith.science/pith/TCRRUWNEACRO6SSABLFPBCOA34/action/citation_signature","submit_replication":"https://pith.science/pith/TCRRUWNEACRO6SSABLFPBCOA34/action/replication_record"}},"created_at":"2026-07-05T01:34:44.550095+00:00","updated_at":"2026-07-05T01:34:44.550095+00:00"}