{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:QMEVTYD6LOOKT5CEQECUPOXRTM","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":"5ee589508547da83a0f126a40c2961a2c51e52f9a139b827cc869930de2bf7a0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2015-08-26T12:07:08Z","title_canon_sha256":"b1f7a992000166feca4339d761d501c4e0b1ebd446e648f13cba1a4c771236d3"},"schema_version":"1.0","source":{"id":"1508.06458","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.06458","created_at":"2026-05-18T01:19:00Z"},{"alias_kind":"arxiv_version","alias_value":"1508.06458v1","created_at":"2026-05-18T01:19:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.06458","created_at":"2026-05-18T01:19:00Z"},{"alias_kind":"pith_short_12","alias_value":"QMEVTYD6LOOK","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"QMEVTYD6LOOKT5CE","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"QMEVTYD6","created_at":"2026-05-18T12:29:37Z"}],"graph_snapshots":[{"event_id":"sha256:a3621114babfaeaf769423073eb9df52898194158633005f98a87205bb61395f","target":"graph","created_at":"2026-05-18T01:19:00Z","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":"In this note we give a necessary condition for having an almost complex structure on the product $S^{2m} \\times M$, where $M$ is a connected orientable closed manifold. We show that if the Euler characteristic $\\chi(M) \\neq 0$, then except for finitely many values of $m$, we do not have almost complex structure on $S^{2m} \\times M$. In the particular case when $M = \\mathbb{C}\\mathbb P^n, n \\neq 1$, we show that if $n \\not \\equiv 3 \\pmod 4$ then $S^{2m} \\times \\mathbb C \\mathbb P^{n}$ has an almost complex structure if and only if $m = 1,3$. As an application we obtain conditions on the nonexis","authors_text":"Ajay Singh Thakur, Prateep Chakraborty","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2015-08-26T12:07:08Z","title":"Nonexistence of Almost Complex Structures on the product $S^{2m} \\times M$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.06458","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:bc5ebb1e0b1982eb6e6aeaed39c49d006cfd7f58fd5d0da2e0658fe00c56bcbf","target":"record","created_at":"2026-05-18T01:19:00Z","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":"5ee589508547da83a0f126a40c2961a2c51e52f9a139b827cc869930de2bf7a0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2015-08-26T12:07:08Z","title_canon_sha256":"b1f7a992000166feca4339d761d501c4e0b1ebd446e648f13cba1a4c771236d3"},"schema_version":"1.0","source":{"id":"1508.06458","kind":"arxiv","version":1}},"canonical_sha256":"830959e07e5b9ca9f444810547baf19b2c58542e337a6eb4599f6bd7ec26a4f7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"830959e07e5b9ca9f444810547baf19b2c58542e337a6eb4599f6bd7ec26a4f7","first_computed_at":"2026-05-18T01:19:00.337950Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:19:00.337950Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"STdmGze/04lQPK1hcR7akJ9fhs3968YGe1ips2J027Es3uDZnrPYj1nTzekqd0B3pwM9ZUbZ//BJ+gz4YjE/Dg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:19:00.338553Z","signed_message":"canonical_sha256_bytes"},"source_id":"1508.06458","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bc5ebb1e0b1982eb6e6aeaed39c49d006cfd7f58fd5d0da2e0658fe00c56bcbf","sha256:a3621114babfaeaf769423073eb9df52898194158633005f98a87205bb61395f"],"state_sha256":"23cee56d5c0a76e759316c2e49aed81a0c2bd591d794530c092a39de1f9a1f32"}