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If $\\pi=\\pi_1(M)\\not=1$ there are at most two such bundle spaces with given action $u:\\pi\\to{Aut}(\\pi_2(M))$. The bundle space has the geometry $\\mathbb{S}^2\\times\\mathbb{E}^2$ (if $\\chi(M)=0$) or $\\mathbb{S}^2\\times\\mathbb{H}^2$ (if $\\chi(M)<0$), except when $B$ is orientable and $\\pi$ is generated by involutions, in which case the act"},"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":"1008.4186","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-08-25T01:58:25Z","cross_cats_sorted":[],"title_canon_sha256":"09b954b46694096678745b74fcdaa41d7ca1ee2740e672725f2b2802eafc871a","abstract_canon_sha256":"7b91754f8573dd93736e9e487a130ae093eb3ce8ae55da636348714ec88e5e6e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:28:43.194492Z","signature_b64":"/Ytk4ZIyUxrf0Um6sd7TxZ9rfT74spNvV5pc/LJIonuz0eC1WlEhe4KTbgYGmCxWj26ZZrRfz1zyva9Ot9M9Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"21a90bec79f9032bdde3ed70b042b598596cd48f42828e7b3680d0ee59e5f662","last_reissued_at":"2026-05-18T03:28:43.193765Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:28:43.193765Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$S^2$-bundles over 2-orbifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Jonathan A. 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The bundle space has the geometry $\\mathbb{S}^2\\times\\mathbb{E}^2$ (if $\\chi(M)=0$) or $\\mathbb{S}^2\\times\\mathbb{H}^2$ (if $\\chi(M)<0$), except when $B$ is orientable and $\\pi$ is generated by involutions, in which case the act"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.4186","kind":"arxiv","version":3},"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":"1008.4186","created_at":"2026-05-18T03:28:43.193868+00:00"},{"alias_kind":"arxiv_version","alias_value":"1008.4186v3","created_at":"2026-05-18T03:28:43.193868+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.4186","created_at":"2026-05-18T03:28:43.193868+00:00"},{"alias_kind":"pith_short_12","alias_value":"EGUQX3DZ7EBS","created_at":"2026-05-18T12:26:06.534383+00:00"},{"alias_kind":"pith_short_16","alias_value":"EGUQX3DZ7EBSXXPD","created_at":"2026-05-18T12:26:06.534383+00:00"},{"alias_kind":"pith_short_8","alias_value":"EGUQX3DZ","created_at":"2026-05-18T12:26:06.534383+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/EGUQX3DZ7EBSXXPD5VYLAQVVTB","json":"https://pith.science/pith/EGUQX3DZ7EBSXXPD5VYLAQVVTB.json","graph_json":"https://pith.science/api/pith-number/EGUQX3DZ7EBSXXPD5VYLAQVVTB/graph.json","events_json":"https://pith.science/api/pith-number/EGUQX3DZ7EBSXXPD5VYLAQVVTB/events.json","paper":"https://pith.science/paper/EGUQX3DZ"},"agent_actions":{"view_html":"https://pith.science/pith/EGUQX3DZ7EBSXXPD5VYLAQVVTB","download_json":"https://pith.science/pith/EGUQX3DZ7EBSXXPD5VYLAQVVTB.json","view_paper":"https://pith.science/paper/EGUQX3DZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1008.4186&json=true","fetch_graph":"https://pith.science/api/pith-number/EGUQX3DZ7EBSXXPD5VYLAQVVTB/graph.json","fetch_events":"https://pith.science/api/pith-number/EGUQX3DZ7EBSXXPD5VYLAQVVTB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EGUQX3DZ7EBSXXPD5VYLAQVVTB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EGUQX3DZ7EBSXXPD5VYLAQVVTB/action/storage_attestation","attest_author":"https://pith.science/pith/EGUQX3DZ7EBSXXPD5VYLAQVVTB/action/author_attestation","sign_citation":"https://pith.science/pith/EGUQX3DZ7EBSXXPD5VYLAQVVTB/action/citation_signature","submit_replication":"https://pith.science/pith/EGUQX3DZ7EBSXXPD5VYLAQVVTB/action/replication_record"}},"created_at":"2026-05-18T03:28:43.193868+00:00","updated_at":"2026-05-18T03:28:43.193868+00:00"}