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Motivated by the $p$--local homotopy classification of these gauge groups, due to Kishimoto--Theriault--Tsutaya and Kameko, we study the low-degree mod~$2$ cohomology of the classifying spaces $B\\mathcal{G}_k$ as unstable modules over the Steenrod algebra. Using the evaluation fibration \\[ \\Omega^3_0G_2\\longrightarrow B\\mathcal{G}_k \\xrightarrow{\\;\\mathrm{ev}\\;} BG_2 \\] and its Serre spectral sequence, we analyze \\[ H^s(BG_2;H^t(\\Omega^3_0G_2;\\mathbb F_2)) \\Long"},"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":"2512.12195","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2025-12-13T05:50:49Z","cross_cats_sorted":[],"title_canon_sha256":"1d30baa755f3ac457a3868d81516e8b34e32d2c9a431f31a6bc514c977da1c82","abstract_canon_sha256":"425a2f5daab7622fbec2624ac9a75f7ffc7887ede52df579d6a13113e83e95b0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-29T02:05:38.361014Z","signature_b64":"0mbe1Ng3r5JyTAZNGyzfWZ9WZQarSBJXXy8L1xksOn4uTEH7ZHu6YXCJKJx9oxxZ9d2N7WqUn3mmM+UaTRxGAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7bc7339049759f49421c5e39e0f8749bd385d0c80ae6a0ce5f4078072d7e19ee","last_reissued_at":"2026-05-29T02:05:38.360517Z","signature_status":"signed_v1","first_computed_at":"2026-05-29T02:05:38.360517Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Low-degree mod 2 cohomology of classifying spaces of $G_2$-gauge groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Dang Vo Phuc","submitted_at":"2025-12-13T05:50:49Z","abstract_excerpt":"Let $\\mathcal{G}_k$ denote the gauge group of the principal $G_2$--bundle over $S^4$ classified by $k\\in \\pi_4(BG_2)\\cong \\mathbb Z$. Motivated by the $p$--local homotopy classification of these gauge groups, due to Kishimoto--Theriault--Tsutaya and Kameko, we study the low-degree mod~$2$ cohomology of the classifying spaces $B\\mathcal{G}_k$ as unstable modules over the Steenrod algebra. Using the evaluation fibration \\[ \\Omega^3_0G_2\\longrightarrow B\\mathcal{G}_k \\xrightarrow{\\;\\mathrm{ev}\\;} BG_2 \\] and its Serre spectral sequence, we analyze \\[ H^s(BG_2;H^t(\\Omega^3_0G_2;\\mathbb F_2)) \\Long"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2512.12195","kind":"arxiv","version":4},"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/2512.12195/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":"2512.12195","created_at":"2026-05-29T02:05:38.360576+00:00"},{"alias_kind":"arxiv_version","alias_value":"2512.12195v4","created_at":"2026-05-29T02:05:38.360576+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2512.12195","created_at":"2026-05-29T02:05:38.360576+00:00"},{"alias_kind":"pith_short_12","alias_value":"PPDTHECJOWPU","created_at":"2026-05-29T02:05:38.360576+00:00"},{"alias_kind":"pith_short_16","alias_value":"PPDTHECJOWPUSQQ4","created_at":"2026-05-29T02:05:38.360576+00:00"},{"alias_kind":"pith_short_8","alias_value":"PPDTHECJ","created_at":"2026-05-29T02:05:38.360576+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/PPDTHECJOWPUSQQ4LY46B6DUTP","json":"https://pith.science/pith/PPDTHECJOWPUSQQ4LY46B6DUTP.json","graph_json":"https://pith.science/api/pith-number/PPDTHECJOWPUSQQ4LY46B6DUTP/graph.json","events_json":"https://pith.science/api/pith-number/PPDTHECJOWPUSQQ4LY46B6DUTP/events.json","paper":"https://pith.science/paper/PPDTHECJ"},"agent_actions":{"view_html":"https://pith.science/pith/PPDTHECJOWPUSQQ4LY46B6DUTP","download_json":"https://pith.science/pith/PPDTHECJOWPUSQQ4LY46B6DUTP.json","view_paper":"https://pith.science/paper/PPDTHECJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2512.12195&json=true","fetch_graph":"https://pith.science/api/pith-number/PPDTHECJOWPUSQQ4LY46B6DUTP/graph.json","fetch_events":"https://pith.science/api/pith-number/PPDTHECJOWPUSQQ4LY46B6DUTP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PPDTHECJOWPUSQQ4LY46B6DUTP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PPDTHECJOWPUSQQ4LY46B6DUTP/action/storage_attestation","attest_author":"https://pith.science/pith/PPDTHECJOWPUSQQ4LY46B6DUTP/action/author_attestation","sign_citation":"https://pith.science/pith/PPDTHECJOWPUSQQ4LY46B6DUTP/action/citation_signature","submit_replication":"https://pith.science/pith/PPDTHECJOWPUSQQ4LY46B6DUTP/action/replication_record"}},"created_at":"2026-05-29T02:05:38.360576+00:00","updated_at":"2026-05-29T02:05:38.360576+00:00"}