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For any class $a\\in H^*(LM)$ of positive degree, we prove that the cup product $\\chi(M)a\\cup ev^*(\\omega)$ is null. In particular, if $i^*:H^*(LM;\\mathbb{F}_p)\\twoheadrightarrow H^*(\\Omega M;\\mathbb{F}_p)$ is onto then $\\chi(M)$ is divisible by $p$ (or $M$ is a point)."},"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":"1308.6684","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2013-08-30T09:02:59Z","cross_cats_sorted":[],"title_canon_sha256":"942222b528010f00731d6f1b9b487c81bbf1f807660c1e1fdf3164cf17de87c0","abstract_canon_sha256":"e7cb6187147e46fbda96576b57dc71ca0d96358f4d483e9bed271d020f8fc49c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:14:38.418117Z","signature_b64":"I92HCo8QETzN7sdfq7UROnQoW0pXUx76nn3eKsqrf9Ixw3eRBZ99rZPbPiYTWDoNlYvJYl6Cghx2XRzwTpcOAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5e574ff0354894d474b6acce974cd85bb690c57a27ee11eac28e346b7c420144","last_reissued_at":"2026-05-18T03:14:38.417620Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:14:38.417620Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"String Topology, Euler Class and TNCZ free loop fibrations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Luc Menichi (LAREMA)","submitted_at":"2013-08-30T09:02:59Z","abstract_excerpt":"Let $M$ be a connected, closed oriented manifold. Let $\\omega\\in H^m(M)$ be its orientation class. Let $\\chi(M)$ be its Euler characteristic. Consider the free loop fibration $\\Omega M\\buildrel{i}\\over\\hookrightarrow LM\\buildrel{ev}\\over\\twoheadrightarrow M$. For any class $a\\in H^*(LM)$ of positive degree, we prove that the cup product $\\chi(M)a\\cup ev^*(\\omega)$ is null. In particular, if $i^*:H^*(LM;\\mathbb{F}_p)\\twoheadrightarrow H^*(\\Omega M;\\mathbb{F}_p)$ is onto then $\\chi(M)$ is divisible by $p$ (or $M$ is a point)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.6684","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1308.6684","created_at":"2026-05-18T03:14:38.417690+00:00"},{"alias_kind":"arxiv_version","alias_value":"1308.6684v1","created_at":"2026-05-18T03:14:38.417690+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.6684","created_at":"2026-05-18T03:14:38.417690+00:00"},{"alias_kind":"pith_short_12","alias_value":"LZLU74BVJCKN","created_at":"2026-05-18T12:27:51.066281+00:00"},{"alias_kind":"pith_short_16","alias_value":"LZLU74BVJCKNI5FW","created_at":"2026-05-18T12:27:51.066281+00:00"},{"alias_kind":"pith_short_8","alias_value":"LZLU74BV","created_at":"2026-05-18T12:27:51.066281+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/LZLU74BVJCKNI5FWVTHJOTGYLO","json":"https://pith.science/pith/LZLU74BVJCKNI5FWVTHJOTGYLO.json","graph_json":"https://pith.science/api/pith-number/LZLU74BVJCKNI5FWVTHJOTGYLO/graph.json","events_json":"https://pith.science/api/pith-number/LZLU74BVJCKNI5FWVTHJOTGYLO/events.json","paper":"https://pith.science/paper/LZLU74BV"},"agent_actions":{"view_html":"https://pith.science/pith/LZLU74BVJCKNI5FWVTHJOTGYLO","download_json":"https://pith.science/pith/LZLU74BVJCKNI5FWVTHJOTGYLO.json","view_paper":"https://pith.science/paper/LZLU74BV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1308.6684&json=true","fetch_graph":"https://pith.science/api/pith-number/LZLU74BVJCKNI5FWVTHJOTGYLO/graph.json","fetch_events":"https://pith.science/api/pith-number/LZLU74BVJCKNI5FWVTHJOTGYLO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LZLU74BVJCKNI5FWVTHJOTGYLO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LZLU74BVJCKNI5FWVTHJOTGYLO/action/storage_attestation","attest_author":"https://pith.science/pith/LZLU74BVJCKNI5FWVTHJOTGYLO/action/author_attestation","sign_citation":"https://pith.science/pith/LZLU74BVJCKNI5FWVTHJOTGYLO/action/citation_signature","submit_replication":"https://pith.science/pith/LZLU74BVJCKNI5FWVTHJOTGYLO/action/replication_record"}},"created_at":"2026-05-18T03:14:38.417690+00:00","updated_at":"2026-05-18T03:14:38.417690+00:00"}