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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 $H^2(L_{-1,n};Z) = Z$."},"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":"0905.2116","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2009-05-13T14:07:05Z","cross_cats_sorted":[],"title_canon_sha256":"4ec34201e19cb12ccc539719349d20aa85fd535723f22be5da1d66b0890d0391","abstract_canon_sha256":"6f2b53cc14a6d965dc27fd740751fc1592e018d458cef77924b4e474c8eafa0f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:00:01.938546Z","signature_b64":"euNGqXgyb8Pjnv0yHZy5Ec98HYxuLJTLX7fdq9aTacK0GnZAYaWxxCPZPa6M8u3U9UCFkSZYOScCMYGc7/WtAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7cc05396c8c76208c3fb8fb291b03c85f29fba33fb39d550d5d8ea90aa42dbc5","last_reissued_at":"2026-05-18T04:00:01.937858Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:00:01.937858Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The homotopy and cohomology of spaces of locally convex curves in the sphere -- II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Nicolau C. Saldanha","submitted_at":"2009-05-13T14:07:05Z","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 $H^2(L_{-1,n};Z) = Z$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0905.2116","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":"0905.2116","created_at":"2026-05-18T04:00:01.937952+00:00"},{"alias_kind":"arxiv_version","alias_value":"0905.2116v2","created_at":"2026-05-18T04:00:01.937952+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0905.2116","created_at":"2026-05-18T04:00:01.937952+00:00"},{"alias_kind":"pith_short_12","alias_value":"PTAFHFWIY5RA","created_at":"2026-05-18T12:26:01.383474+00:00"},{"alias_kind":"pith_short_16","alias_value":"PTAFHFWIY5RARQ73","created_at":"2026-05-18T12:26:01.383474+00:00"},{"alias_kind":"pith_short_8","alias_value":"PTAFHFWI","created_at":"2026-05-18T12:26:01.383474+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2002.03986","citing_title":"Characterization of some convex curves on the 3-sphere","ref_index":20,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/PTAFHFWIY5RARQ73R6ZJDMB4QX","json":"https://pith.science/pith/PTAFHFWIY5RARQ73R6ZJDMB4QX.json","graph_json":"https://pith.science/api/pith-number/PTAFHFWIY5RARQ73R6ZJDMB4QX/graph.json","events_json":"https://pith.science/api/pith-number/PTAFHFWIY5RARQ73R6ZJDMB4QX/events.json","paper":"https://pith.science/paper/PTAFHFWI"},"agent_actions":{"view_html":"https://pith.science/pith/PTAFHFWIY5RARQ73R6ZJDMB4QX","download_json":"https://pith.science/pith/PTAFHFWIY5RARQ73R6ZJDMB4QX.json","view_paper":"https://pith.science/paper/PTAFHFWI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0905.2116&json=true","fetch_graph":"https://pith.science/api/pith-number/PTAFHFWIY5RARQ73R6ZJDMB4QX/graph.json","fetch_events":"https://pith.science/api/pith-number/PTAFHFWIY5RARQ73R6ZJDMB4QX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PTAFHFWIY5RARQ73R6ZJDMB4QX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PTAFHFWIY5RARQ73R6ZJDMB4QX/action/storage_attestation","attest_author":"https://pith.science/pith/PTAFHFWIY5RARQ73R6ZJDMB4QX/action/author_attestation","sign_citation":"https://pith.science/pith/PTAFHFWIY5RARQ73R6ZJDMB4QX/action/citation_signature","submit_replication":"https://pith.science/pith/PTAFHFWIY5RARQ73R6ZJDMB4QX/action/replication_record"}},"created_at":"2026-05-18T04:00:01.937952+00:00","updated_at":"2026-05-18T04:00:01.937952+00:00"}