{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:WEGMVO36F7KL4V64GAKDDXXJZR","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":"afa0a9f485eab02f6338120202a4c8c5f5afa8f31596c19f014a61537c942002","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2013-09-25T14:05:55Z","title_canon_sha256":"abe1752d24e9f7bb5c72b715d4ceadf1192ecc96d3597cd8b1dc4a0cf22332e1"},"schema_version":"1.0","source":{"id":"1309.6510","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.6510","created_at":"2026-05-18T03:12:20Z"},{"alias_kind":"arxiv_version","alias_value":"1309.6510v1","created_at":"2026-05-18T03:12:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.6510","created_at":"2026-05-18T03:12:20Z"},{"alias_kind":"pith_short_12","alias_value":"WEGMVO36F7KL","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_16","alias_value":"WEGMVO36F7KL4V64","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_8","alias_value":"WEGMVO36","created_at":"2026-05-18T12:28:04Z"}],"graph_snapshots":[{"event_id":"sha256:cc1eb43fa3555c34b3fe5414b71a2c59b8feb4806f45c2bb47eaa088cb20bdb9","target":"graph","created_at":"2026-05-18T03:12:20Z","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":"It is well-known how to compute the structure of the second homotopy group of a space, $X$, as a module over the fundamental group, $\\pi_1X$, using the homology of the universal cover and the Hurewicz isomorphism. We describe a new method to compute the third homotopy group, $\\pi_3 X$, as a module over $\\pi_1 X$. Moreover, we determine $\\pi_3 X$ as an extension of $\\pi_1 X$-modules derived from Whitehead's Certain Exact Sequence. Our method is based on the theory of quadratic modules. Explicit computations are carried out for pseudo-projective 3-spaces $X = S^1 \\cup e^2 \\cup e^3$ consisting of","authors_text":"Beatrice Bleile, Hans-Joachim Baues","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2013-09-25T14:05:55Z","title":"The Third Homotopy Group as a pi_1-Module"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.6510","kind":"arxiv","version":1},"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":"sha256:505f85497ab090e0c19460b3275e35027aaa15d61f356273110a0b9ef223e319","target":"record","created_at":"2026-05-18T03:12:20Z","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":"afa0a9f485eab02f6338120202a4c8c5f5afa8f31596c19f014a61537c942002","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2013-09-25T14:05:55Z","title_canon_sha256":"abe1752d24e9f7bb5c72b715d4ceadf1192ecc96d3597cd8b1dc4a0cf22332e1"},"schema_version":"1.0","source":{"id":"1309.6510","kind":"arxiv","version":1}},"canonical_sha256":"b10ccabb7e2fd4be57dc301431dee9cc5243865a875c6222b90877ffde3deab1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b10ccabb7e2fd4be57dc301431dee9cc5243865a875c6222b90877ffde3deab1","first_computed_at":"2026-05-18T03:12:20.810068Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:12:20.810068Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WuTDbesgMWRkS8mHYNktvVUZ39LLhJ/W1+vf3D2NjfvrnLsNDLgr4rNgLFtZePtVHO/5cVFH8HdHuzWvuVIwAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:12:20.810965Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.6510","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:505f85497ab090e0c19460b3275e35027aaa15d61f356273110a0b9ef223e319","sha256:cc1eb43fa3555c34b3fe5414b71a2c59b8feb4806f45c2bb47eaa088cb20bdb9"],"state_sha256":"d8b9962fd9642bcf89131b0921d8d7799e669f91ace823b3d156b487b57eee53"}