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The algebra $P_k$ is a module over the mod-2 Steenrod algebra $\\mathcal A$. In 1989, Singer [22] defined the $k$-th homological algebraic transfer, which is a homomorphism $$\\varphi_k=(\\varphi_k)_m :{\\rm Tor}^{\\mathcal A}_{k,k+m} (\\mathbb F_2,\\mathbb F_2) \\to (\\mathbb F_2\\otimes_{\\mathcal A}P_k)_m^{GL_k}$$ from the ho"},"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":"2505.23218","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AT","submitted_at":"2025-05-29T08:02:44Z","cross_cats_sorted":[],"title_canon_sha256":"14fdbc2d9aa50fba7cbbb2217c35177dccc994eb563d61aadbef3d83c9194d3f","abstract_canon_sha256":"f3befd84deda443060379ac37e0a9dfa258a42133b7e0d65a181085296d118c1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:11:57.675490Z","signature_b64":"3rDACoDQeJQWe8MA72qb4Wp7N+bV/7HitjmoCPPHPZSCmifbjkGBNX+79DQFzhHru2HhkXPPJ1XSYcAKerzXBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d806bd173a2b4d6bc3918021d97dc59fca2099df87bec4a7ae12846f65b7cb13","last_reissued_at":"2026-07-05T11:11:57.674923Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:11:57.674923Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An application of the hit problem to the algebraic transfer","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Nguyen Sum","submitted_at":"2025-05-29T08:02:44Z","abstract_excerpt":"Let $P_k$ be the polynomial algebra $\\mathbb F_2[x_1,x_2,\\ldots ,x_k]$ over the field $\\mathbb F_2$ with two elements, in $k$ variables $x_1, x_2, \\ldots , x_k$, each variable of degree 1. Denote by $GL_k$ the general linear group over $\\mathbb F_2$ which regularly acts on $P_k$. The algebra $P_k$ is a module over the mod-2 Steenrod algebra $\\mathcal A$. In 1989, Singer [22] defined the $k$-th homological algebraic transfer, which is a homomorphism $$\\varphi_k=(\\varphi_k)_m :{\\rm Tor}^{\\mathcal A}_{k,k+m} (\\mathbb F_2,\\mathbb F_2) \\to (\\mathbb F_2\\otimes_{\\mathcal A}P_k)_m^{GL_k}$$ from the ho"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.23218","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2505.23218/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2505.23218","created_at":"2026-07-05T11:11:57.674983+00:00"},{"alias_kind":"arxiv_version","alias_value":"2505.23218v1","created_at":"2026-07-05T11:11:57.674983+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2505.23218","created_at":"2026-07-05T11:11:57.674983+00:00"},{"alias_kind":"pith_short_12","alias_value":"3ADL2FZ2FNGW","created_at":"2026-07-05T11:11:57.674983+00:00"},{"alias_kind":"pith_short_16","alias_value":"3ADL2FZ2FNGWXQ4R","created_at":"2026-07-05T11:11:57.674983+00:00"},{"alias_kind":"pith_short_8","alias_value":"3ADL2FZ2","created_at":"2026-07-05T11:11:57.674983+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/3ADL2FZ2FNGWXQ4RQAQ5S7OFT7","json":"https://pith.science/pith/3ADL2FZ2FNGWXQ4RQAQ5S7OFT7.json","graph_json":"https://pith.science/api/pith-number/3ADL2FZ2FNGWXQ4RQAQ5S7OFT7/graph.json","events_json":"https://pith.science/api/pith-number/3ADL2FZ2FNGWXQ4RQAQ5S7OFT7/events.json","paper":"https://pith.science/paper/3ADL2FZ2"},"agent_actions":{"view_html":"https://pith.science/pith/3ADL2FZ2FNGWXQ4RQAQ5S7OFT7","download_json":"https://pith.science/pith/3ADL2FZ2FNGWXQ4RQAQ5S7OFT7.json","view_paper":"https://pith.science/paper/3ADL2FZ2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2505.23218&json=true","fetch_graph":"https://pith.science/api/pith-number/3ADL2FZ2FNGWXQ4RQAQ5S7OFT7/graph.json","fetch_events":"https://pith.science/api/pith-number/3ADL2FZ2FNGWXQ4RQAQ5S7OFT7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3ADL2FZ2FNGWXQ4RQAQ5S7OFT7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3ADL2FZ2FNGWXQ4RQAQ5S7OFT7/action/storage_attestation","attest_author":"https://pith.science/pith/3ADL2FZ2FNGWXQ4RQAQ5S7OFT7/action/author_attestation","sign_citation":"https://pith.science/pith/3ADL2FZ2FNGWXQ4RQAQ5S7OFT7/action/citation_signature","submit_replication":"https://pith.science/pith/3ADL2FZ2FNGWXQ4RQAQ5S7OFT7/action/replication_record"}},"created_at":"2026-07-05T11:11:57.674983+00:00","updated_at":"2026-07-05T11:11:57.674983+00:00"}