{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:1996:YKL2GF5GV722HR264QYHHJEBWQ","short_pith_number":"pith:YKL2GF5G","schema_version":"1.0","canonical_sha256":"c297a317a6aff5a3c75ee43073a481b40d1ede6e222432f958db1e0a783a8e80","source":{"kind":"arxiv","id":"dg-ga/9612004","version":1},"attestation_state":"computed","paper":{"title":"Circle-valued Morse theory, Reidemeister torsion, and Seiberg-Witten invariants of 3-manifolds","license":"","headline":"","cross_cats":["math.DG"],"primary_cat":"dg-ga","authors_text":"Michael Hutchings, Yi-Jen Lee","submitted_at":"1996-12-03T17:58:11Z","abstract_excerpt":"Let X be a compact oriented Riemannian manifold and let $\\phi:X\\to S^1$ be a circle-valued Morse function. Under some mild assumptions on $\\phi$, we prove a formula relating:\n  (a) the number of closed orbits of the gradient flow of $\\phi$ of any given degree;\n  (b) the torsion of a ``Morse complex'', which counts gradient flow lines between critical points of $\\phi$; and\n  (c) a kind of Reidemeister torsion of X determined by the homotopy class of $\\phi$.\n  When $\\dim(X)=3$ and $b_1(X)>0$, we state a conjecture analogous to Taubes's ``SW=Gromov'' theorem, and we use it to deduce (for closed 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":"dg-ga/9612004","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"dg-ga","submitted_at":"1996-12-03T17:58:11Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"151cf67d9a4a4090664001cc7356d43be491f6a8f87a7fcf4921fa8693242809","abstract_canon_sha256":"f4a41d2fb9368de995ca64a672aeac2432bac3e683f573322c779b029a756578"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:06:48.548921Z","signature_b64":"q9hu7vgsGJWJmSfD4WoIH0TRdy/7ZSt0ztnLsAHNmGKa1D7TfmWvzTBoOOlw5bl4oxMKzAHS3Gca+NWHume7Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c297a317a6aff5a3c75ee43073a481b40d1ede6e222432f958db1e0a783a8e80","last_reissued_at":"2026-05-18T01:06:48.548256Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:06:48.548256Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Circle-valued Morse theory, Reidemeister torsion, and Seiberg-Witten invariants of 3-manifolds","license":"","headline":"","cross_cats":["math.DG"],"primary_cat":"dg-ga","authors_text":"Michael Hutchings, Yi-Jen Lee","submitted_at":"1996-12-03T17:58:11Z","abstract_excerpt":"Let X be a compact oriented Riemannian manifold and let $\\phi:X\\to S^1$ be a circle-valued Morse function. Under some mild assumptions on $\\phi$, we prove a formula relating:\n  (a) the number of closed orbits of the gradient flow of $\\phi$ of any given degree;\n  (b) the torsion of a ``Morse complex'', which counts gradient flow lines between critical points of $\\phi$; and\n  (c) a kind of Reidemeister torsion of X determined by the homotopy class of $\\phi$.\n  When $\\dim(X)=3$ and $b_1(X)>0$, we state a conjecture analogous to Taubes's ``SW=Gromov'' theorem, and we use it to deduce (for closed m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"dg-ga/9612004","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":"dg-ga/9612004","created_at":"2026-05-18T01:06:48.548359+00:00"},{"alias_kind":"arxiv_version","alias_value":"dg-ga/9612004v1","created_at":"2026-05-18T01:06:48.548359+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.dg-ga/9612004","created_at":"2026-05-18T01:06:48.548359+00:00"},{"alias_kind":"pith_short_12","alias_value":"YKL2GF5GV722","created_at":"2026-05-18T12:25:48.327863+00:00"},{"alias_kind":"pith_short_16","alias_value":"YKL2GF5GV722HR26","created_at":"2026-05-18T12:25:48.327863+00:00"},{"alias_kind":"pith_short_8","alias_value":"YKL2GF5G","created_at":"2026-05-18T12:25:48.327863+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/YKL2GF5GV722HR264QYHHJEBWQ","json":"https://pith.science/pith/YKL2GF5GV722HR264QYHHJEBWQ.json","graph_json":"https://pith.science/api/pith-number/YKL2GF5GV722HR264QYHHJEBWQ/graph.json","events_json":"https://pith.science/api/pith-number/YKL2GF5GV722HR264QYHHJEBWQ/events.json","paper":"https://pith.science/paper/YKL2GF5G"},"agent_actions":{"view_html":"https://pith.science/pith/YKL2GF5GV722HR264QYHHJEBWQ","download_json":"https://pith.science/pith/YKL2GF5GV722HR264QYHHJEBWQ.json","view_paper":"https://pith.science/paper/YKL2GF5G","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=dg-ga/9612004&json=true","fetch_graph":"https://pith.science/api/pith-number/YKL2GF5GV722HR264QYHHJEBWQ/graph.json","fetch_events":"https://pith.science/api/pith-number/YKL2GF5GV722HR264QYHHJEBWQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YKL2GF5GV722HR264QYHHJEBWQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YKL2GF5GV722HR264QYHHJEBWQ/action/storage_attestation","attest_author":"https://pith.science/pith/YKL2GF5GV722HR264QYHHJEBWQ/action/author_attestation","sign_citation":"https://pith.science/pith/YKL2GF5GV722HR264QYHHJEBWQ/action/citation_signature","submit_replication":"https://pith.science/pith/YKL2GF5GV722HR264QYHHJEBWQ/action/replication_record"}},"created_at":"2026-05-18T01:06:48.548359+00:00","updated_at":"2026-05-18T01:06:48.548359+00:00"}