{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:GYCFP3Z37HLOD36FONPSWU5C5Z","short_pith_number":"pith:GYCFP3Z3","schema_version":"1.0","canonical_sha256":"360457ef3bf9d6e1efc5735f2b53a2ee4bd72d3bff943bec0a556bcb7f8b597b","source":{"kind":"arxiv","id":"1811.01706","version":2},"attestation_state":"computed","paper":{"title":"Estimates by gap potentials of free homotopy decompositions of critical Sobolev maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.AT"],"primary_cat":"math.FA","authors_text":"Jean Van Schaftingen (UCLouvain)","submitted_at":"2018-11-05T14:22:06Z","abstract_excerpt":"A free homotopy decomposition of any continuous map from a compact Riemmanian manifold $\\mathcal{M}$ to a compact Riemannian manifold $\\mathcal{N}$ into a finite number maps belonging to a finite set is constructed, in such a way that the number of maps in this free homotopy decomposition and the number of elements of the set to which they belong can be estimated a priori by the critical Sobolev energy of the map in $W^{s,p} (\\mathcal{M}, \\mathcal{N})$, with $sp = m = \\dim \\mathcal{M}$. In particular, when the fundamental group $\\pi_1 (\\mathcal{N})$ acts trivially on the homotopy group $\\pi_m "},"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":"1811.01706","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-11-05T14:22:06Z","cross_cats_sorted":["math.AP","math.AT"],"title_canon_sha256":"9e9fef48b2e4061ad2a40aacdbaffb25268a22a585759810172688c4ea8a1002","abstract_canon_sha256":"f766101a7335164cb2d02e64c59a1e0f32524fb30de414d3fa33cf30729c6f36"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:47:15.767046Z","signature_b64":"wBa8159dvyKT/kIRHF5T23yOccUf9yzVvTXj8GE4+3tBMvB7oXWCBBs/Jd0OrwTtQAdeuc1z9AcqqDMKnqcaAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"360457ef3bf9d6e1efc5735f2b53a2ee4bd72d3bff943bec0a556bcb7f8b597b","last_reissued_at":"2026-07-05T00:47:15.766580Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:47:15.766580Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Estimates by gap potentials of free homotopy decompositions of critical Sobolev maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.AT"],"primary_cat":"math.FA","authors_text":"Jean Van Schaftingen (UCLouvain)","submitted_at":"2018-11-05T14:22:06Z","abstract_excerpt":"A free homotopy decomposition of any continuous map from a compact Riemmanian manifold $\\mathcal{M}$ to a compact Riemannian manifold $\\mathcal{N}$ into a finite number maps belonging to a finite set is constructed, in such a way that the number of maps in this free homotopy decomposition and the number of elements of the set to which they belong can be estimated a priori by the critical Sobolev energy of the map in $W^{s,p} (\\mathcal{M}, \\mathcal{N})$, with $sp = m = \\dim \\mathcal{M}$. In particular, when the fundamental group $\\pi_1 (\\mathcal{N})$ acts trivially on the homotopy group $\\pi_m "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.01706","kind":"arxiv","version":2},"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/1811.01706/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":"1811.01706","created_at":"2026-07-05T00:47:15.766639+00:00"},{"alias_kind":"arxiv_version","alias_value":"1811.01706v2","created_at":"2026-07-05T00:47:15.766639+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.01706","created_at":"2026-07-05T00:47:15.766639+00:00"},{"alias_kind":"pith_short_12","alias_value":"GYCFP3Z37HLO","created_at":"2026-07-05T00:47:15.766639+00:00"},{"alias_kind":"pith_short_16","alias_value":"GYCFP3Z37HLOD36F","created_at":"2026-07-05T00:47:15.766639+00:00"},{"alias_kind":"pith_short_8","alias_value":"GYCFP3Z3","created_at":"2026-07-05T00:47:15.766639+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"1908.06179","citing_title":"Exponential integrability in the spirit of Moser-Trudinger's inequalities of functions with finite non-local, non-convex energy","ref_index":45,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/GYCFP3Z37HLOD36FONPSWU5C5Z","json":"https://pith.science/pith/GYCFP3Z37HLOD36FONPSWU5C5Z.json","graph_json":"https://pith.science/api/pith-number/GYCFP3Z37HLOD36FONPSWU5C5Z/graph.json","events_json":"https://pith.science/api/pith-number/GYCFP3Z37HLOD36FONPSWU5C5Z/events.json","paper":"https://pith.science/paper/GYCFP3Z3"},"agent_actions":{"view_html":"https://pith.science/pith/GYCFP3Z37HLOD36FONPSWU5C5Z","download_json":"https://pith.science/pith/GYCFP3Z37HLOD36FONPSWU5C5Z.json","view_paper":"https://pith.science/paper/GYCFP3Z3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1811.01706&json=true","fetch_graph":"https://pith.science/api/pith-number/GYCFP3Z37HLOD36FONPSWU5C5Z/graph.json","fetch_events":"https://pith.science/api/pith-number/GYCFP3Z37HLOD36FONPSWU5C5Z/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GYCFP3Z37HLOD36FONPSWU5C5Z/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GYCFP3Z37HLOD36FONPSWU5C5Z/action/storage_attestation","attest_author":"https://pith.science/pith/GYCFP3Z37HLOD36FONPSWU5C5Z/action/author_attestation","sign_citation":"https://pith.science/pith/GYCFP3Z37HLOD36FONPSWU5C5Z/action/citation_signature","submit_replication":"https://pith.science/pith/GYCFP3Z37HLOD36FONPSWU5C5Z/action/replication_record"}},"created_at":"2026-07-05T00:47:15.766639+00:00","updated_at":"2026-07-05T00:47:15.766639+00:00"}