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We set $$\\tau(M,\\chi):= \\sup \\inf \\lambda\\_1^+(g)$$ where the infimum runs over all metrics $g$ of volume 1 in a conformal class $[g\\_0]$ on $M$ and where the supremum runs over all conformal classes $[g\\_0]$ on $M$. Let $(M^#,\\chi^#)$ be obtained from $(M,\\chi)$ by 0-dimensional surgery. We prove that $$\\tau(M^#,\\chi^#)\\geq \\tau(M,\\chi).$$ As a corollary we can calculate $\\tau(M,\\chi)$ for any Riemann surface "},"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":"math/0607716","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2006-07-27T15:42:42Z","cross_cats_sorted":[],"title_canon_sha256":"efe613e96f69287c30bfa5f07984d8ae9fbcd1ee1d00d9d72b35a76c98a82e9b","abstract_canon_sha256":"593a7c5778d1c4c72dfb4ed6cdf7a63c42495a578cdae02891a42a894559df23"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:28:44.720925Z","signature_b64":"zeStJoGZOr6Kg2FnJO/rd/Q32noajZ1UMhgVLV9lAFZRAO7qsU/niEyxD8no0tbOcN+X7dlQhaUrveBvzX7QCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ee8197a1d2a43c2c63b1f1ab6e6ce7d9212cfd9056a511676b769fbb4b3a5601","last_reissued_at":"2026-05-18T01:28:44.720416Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:28:44.720416Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The spinorial \\tau-invariant and 0-dimensional surgery","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Bernd Ammann (IECN), Emmanuel Humbert (IECN)","submitted_at":"2006-07-27T15:42:42Z","abstract_excerpt":"Let $M$ be a compact manifold with a metric $g$ and with a fixed spin structure $\\chi$. Let $\\lambda\\_1^+(g)$ be the first non-negative eigenvalue of the Dirac operator on $(M,g,\\chi)$. We set $$\\tau(M,\\chi):= \\sup \\inf \\lambda\\_1^+(g)$$ where the infimum runs over all metrics $g$ of volume 1 in a conformal class $[g\\_0]$ on $M$ and where the supremum runs over all conformal classes $[g\\_0]$ on $M$. Let $(M^#,\\chi^#)$ be obtained from $(M,\\chi)$ by 0-dimensional surgery. We prove that $$\\tau(M^#,\\chi^#)\\geq \\tau(M,\\chi).$$ As a corollary we can calculate $\\tau(M,\\chi)$ for any Riemann surface "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0607716","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":""},"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":"math/0607716","created_at":"2026-05-18T01:28:44.720497+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0607716v2","created_at":"2026-05-18T01:28:44.720497+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0607716","created_at":"2026-05-18T01:28:44.720497+00:00"},{"alias_kind":"pith_short_12","alias_value":"52AZPIOSUQ6C","created_at":"2026-05-18T12:25:53.939244+00:00"},{"alias_kind":"pith_short_16","alias_value":"52AZPIOSUQ6CYY5R","created_at":"2026-05-18T12:25:53.939244+00:00"},{"alias_kind":"pith_short_8","alias_value":"52AZPIOS","created_at":"2026-05-18T12:25:53.939244+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/52AZPIOSUQ6CYY5R6GVW43HH3E","json":"https://pith.science/pith/52AZPIOSUQ6CYY5R6GVW43HH3E.json","graph_json":"https://pith.science/api/pith-number/52AZPIOSUQ6CYY5R6GVW43HH3E/graph.json","events_json":"https://pith.science/api/pith-number/52AZPIOSUQ6CYY5R6GVW43HH3E/events.json","paper":"https://pith.science/paper/52AZPIOS"},"agent_actions":{"view_html":"https://pith.science/pith/52AZPIOSUQ6CYY5R6GVW43HH3E","download_json":"https://pith.science/pith/52AZPIOSUQ6CYY5R6GVW43HH3E.json","view_paper":"https://pith.science/paper/52AZPIOS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0607716&json=true","fetch_graph":"https://pith.science/api/pith-number/52AZPIOSUQ6CYY5R6GVW43HH3E/graph.json","fetch_events":"https://pith.science/api/pith-number/52AZPIOSUQ6CYY5R6GVW43HH3E/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/52AZPIOSUQ6CYY5R6GVW43HH3E/action/timestamp_anchor","attest_storage":"https://pith.science/pith/52AZPIOSUQ6CYY5R6GVW43HH3E/action/storage_attestation","attest_author":"https://pith.science/pith/52AZPIOSUQ6CYY5R6GVW43HH3E/action/author_attestation","sign_citation":"https://pith.science/pith/52AZPIOSUQ6CYY5R6GVW43HH3E/action/citation_signature","submit_replication":"https://pith.science/pith/52AZPIOSUQ6CYY5R6GVW43HH3E/action/replication_record"}},"created_at":"2026-05-18T01:28:44.720497+00:00","updated_at":"2026-05-18T01:28:44.720497+00:00"}