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We also compute the cohomology rings $H^*(G; A)$ for $A = \\mathbb{Z}$ and $A = \\mathbb{Z}/p$ for an odd prime $p$, and indicate how to compute the groups $H^*(G; A)$ and the multiplicative structure given by the cup product for any system of coefficients $A$."},"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":"1412.3606","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2014-12-11T11:15:50Z","cross_cats_sorted":[],"title_canon_sha256":"2ae2cf05a6f890e005998676e7a390b89d40d4680f01885cf759af2f16fa3019","abstract_canon_sha256":"d240218d0d343e005213ae00e3246a39f8654eae463186b93917315d951402c7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:03:06.625624Z","signature_b64":"dzIOxJWdE0QagFPBdXzfA/4yMXBnADXQWkr4n9GX/WumepSCrg6Rylmq8zTIu7DuGGJLc5Yu8/ZDrXeHV0RgDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3ec398b32d88c414a8584e8d64928ef4da99ae7c92e99da5c8c46b1fbf3c13e0","last_reissued_at":"2026-05-18T01:03:06.625124Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:03:06.625124Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The cohomology ring of the sapphires that admit the Sol geometry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Daciberg Lima Gon\\c{c}alves, S\\'ergio Tadao Martins","submitted_at":"2014-12-11T11:15:50Z","abstract_excerpt":"Let $G$ be the fundamental group of a sapphire that admits the Sol geometry and is not a torus bundle. 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