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We generalize the Pin$(2)$-equivariant family Bauer-Furuta invariant to nonsimply connected manifolds, and construct a refinement of this invariant. We use it to show that, if $X_1,X_2$ are two homology tori such that the determinants $r_1,r_2$ of them are odd, then the Dehn twist along a $3$-sphere in the neck of $X_1\\# X_2$ is not smoothly isotop"},"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":"2410.02461","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2024-10-03T13:11:41Z","cross_cats_sorted":[],"title_canon_sha256":"7a77592c99ae4681146a85eef0c0dff8b932fa50ca96f4b66bee5da8d956a128","abstract_canon_sha256":"4f9e2a8adff25c8bd9042015e1349427f16e30e7567479c791490b1bbc705a2e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:20:15.687366Z","signature_b64":"Rtm5hS55+uD4nUEoMSByxb9fr6Ikebn9vN7tASR0kV7NXB/7SaGit9Cn/ghhdAk+DigNwLD/Lawqp/metW1iAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f25f5c6d79261f5fe54408ebacd59c2117a4555262851ed39e0f118d80b4bbde","last_reissued_at":"2026-07-05T10:20:15.686864Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:20:15.686864Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Dehn twist on a connected sum of two homology tori","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Haochen Qiu","submitted_at":"2024-10-03T13:11:41Z","abstract_excerpt":"Kronheimer-Mrowka shows that the Dehn twist along a $3$-sphere in the neck of two $K3$ surfaces is not smoothly isotopic to the identity. Their result requires that the manifolds are simply connected and the signature of one of them is $16 \\mod 32$. We generalize the Pin$(2)$-equivariant family Bauer-Furuta invariant to nonsimply connected manifolds, and construct a refinement of this invariant. We use it to show that, if $X_1,X_2$ are two homology tori such that the determinants $r_1,r_2$ of them are odd, then the Dehn twist along a $3$-sphere in the neck of $X_1\\# X_2$ is not smoothly isotop"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.02461","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/2410.02461/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":"2410.02461","created_at":"2026-07-05T10:20:15.686922+00:00"},{"alias_kind":"arxiv_version","alias_value":"2410.02461v2","created_at":"2026-07-05T10:20:15.686922+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2410.02461","created_at":"2026-07-05T10:20:15.686922+00:00"},{"alias_kind":"pith_short_12","alias_value":"6JPVY3LZEYPV","created_at":"2026-07-05T10:20:15.686922+00:00"},{"alias_kind":"pith_short_16","alias_value":"6JPVY3LZEYPV7ZKE","created_at":"2026-07-05T10:20:15.686922+00:00"},{"alias_kind":"pith_short_8","alias_value":"6JPVY3LZ","created_at":"2026-07-05T10:20:15.686922+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":2,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2604.25366","citing_title":"A gerbe-like construction in gauge theory II: the case of homology tori","ref_index":12,"is_internal_anchor":false},{"citing_arxiv_id":"2604.15087","citing_title":"Diffeomorphism groups and gauge theory for families","ref_index":76,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/6JPVY3LZEYPV7ZKEBDV2ZVM4EE","json":"https://pith.science/pith/6JPVY3LZEYPV7ZKEBDV2ZVM4EE.json","graph_json":"https://pith.science/api/pith-number/6JPVY3LZEYPV7ZKEBDV2ZVM4EE/graph.json","events_json":"https://pith.science/api/pith-number/6JPVY3LZEYPV7ZKEBDV2ZVM4EE/events.json","paper":"https://pith.science/paper/6JPVY3LZ"},"agent_actions":{"view_html":"https://pith.science/pith/6JPVY3LZEYPV7ZKEBDV2ZVM4EE","download_json":"https://pith.science/pith/6JPVY3LZEYPV7ZKEBDV2ZVM4EE.json","view_paper":"https://pith.science/paper/6JPVY3LZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2410.02461&json=true","fetch_graph":"https://pith.science/api/pith-number/6JPVY3LZEYPV7ZKEBDV2ZVM4EE/graph.json","fetch_events":"https://pith.science/api/pith-number/6JPVY3LZEYPV7ZKEBDV2ZVM4EE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6JPVY3LZEYPV7ZKEBDV2ZVM4EE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6JPVY3LZEYPV7ZKEBDV2ZVM4EE/action/storage_attestation","attest_author":"https://pith.science/pith/6JPVY3LZEYPV7ZKEBDV2ZVM4EE/action/author_attestation","sign_citation":"https://pith.science/pith/6JPVY3LZEYPV7ZKEBDV2ZVM4EE/action/citation_signature","submit_replication":"https://pith.science/pith/6JPVY3LZEYPV7ZKEBDV2ZVM4EE/action/replication_record"}},"created_at":"2026-07-05T10:20:15.686922+00:00","updated_at":"2026-07-05T10:20:15.686922+00:00"}