{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:PTAFHFWIY5RARQ73R6ZJDMB4QX","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"6f2b53cc14a6d965dc27fd740751fc1592e018d458cef77924b4e474c8eafa0f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2009-05-13T14:07:05Z","title_canon_sha256":"4ec34201e19cb12ccc539719349d20aa85fd535723f22be5da1d66b0890d0391"},"schema_version":"1.0","source":{"id":"0905.2116","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0905.2116","created_at":"2026-05-18T04:00:01Z"},{"alias_kind":"arxiv_version","alias_value":"0905.2116v2","created_at":"2026-05-18T04:00:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0905.2116","created_at":"2026-05-18T04:00:01Z"},{"alias_kind":"pith_short_12","alias_value":"PTAFHFWIY5RA","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"PTAFHFWIY5RARQ73","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"PTAFHFWI","created_at":"2026-05-18T12:26:01Z"}],"graph_snapshots":[{"event_id":"sha256:c5aa287ee52ff56c72e83e3c2429c0819197bb7d607c6ed3e149427685351603","target":"graph","created_at":"2026-05-18T04:00:01Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"A smooth curve $\\gamma: [0,1] \\to S^2$ is locally convex if its geodesic curvature is positive at every point. J. A. Little showed that the space of all locally positive curves $\\gamma$ with $\\gamma(0) = \\gamma(1) = e_1$ and $\\gamma'(0) = \\gamma'(1) = e_2$ has three connected components $L_{-1,c}$, $L_{+1}$, $L_{-1,n}$. The space $L_{-1,c}$ is known to be contractible but the topology of the other two connected components is not well understood. We prove that all connected components of $L_I$ are simply connected, that $H^2(L_{+1};Z) = Z^2$ and $H^2(L_{-1,n};Z) = Z$.","authors_text":"Nicolau C. Saldanha","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2009-05-13T14:07:05Z","title":"The homotopy and cohomology of spaces of locally convex curves in the sphere -- II"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0905.2116","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:47506fb21d036400807f7269daae0030e2aa1dfdc2cb9bc15b82ecb76962ed4c","target":"record","created_at":"2026-05-18T04:00:01Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"6f2b53cc14a6d965dc27fd740751fc1592e018d458cef77924b4e474c8eafa0f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2009-05-13T14:07:05Z","title_canon_sha256":"4ec34201e19cb12ccc539719349d20aa85fd535723f22be5da1d66b0890d0391"},"schema_version":"1.0","source":{"id":"0905.2116","kind":"arxiv","version":2}},"canonical_sha256":"7cc05396c8c76208c3fb8fb291b03c85f29fba33fb39d550d5d8ea90aa42dbc5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7cc05396c8c76208c3fb8fb291b03c85f29fba33fb39d550d5d8ea90aa42dbc5","first_computed_at":"2026-05-18T04:00:01.937858Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:00:01.937858Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"euNGqXgyb8Pjnv0yHZy5Ec98HYxuLJTLX7fdq9aTacK0GnZAYaWxxCPZPa6M8u3U9UCFkSZYOScCMYGc7/WtAA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:00:01.938546Z","signed_message":"canonical_sha256_bytes"},"source_id":"0905.2116","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:47506fb21d036400807f7269daae0030e2aa1dfdc2cb9bc15b82ecb76962ed4c","sha256:c5aa287ee52ff56c72e83e3c2429c0819197bb7d607c6ed3e149427685351603"],"state_sha256":"1307ecdfa5986743da19fafdf5936ea29b15a75e5efc2c20fc49dfd61d401157"}