{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:NR4M4I5KVBGDTPFJY2FYL4C5ZI","short_pith_number":"pith:NR4M4I5K","schema_version":"1.0","canonical_sha256":"6c78ce23aaa84c39bca9c68b85f05dca0988533237742bf242d4727fc0ad8d05","source":{"kind":"arxiv","id":"1210.8252","version":2},"attestation_state":"computed","paper":{"title":"Homotopy pullback of $A_n$-spaces and its applications to $A_n$-types of gauge groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Mitsunobu Tsutaya","submitted_at":"2012-10-31T07:37:32Z","abstract_excerpt":"We construct the homotopy pullback of $A_n$-spaces and show some universal property of it. As the first application, we review the Zabrodsky's result which states that for each prime $p$, there is a finite CW complex which admits an $A_{p-1}$-form but no $A_p$-form. As the second application, we investigate $A_n$-types of gauge groups. In particular, we give a new result on $A_n$-types of the gauge groups of principal $\\mathrm{SU}(2)$-bundles over $S^4$, which is a complete classification when they are localized away from 2."},"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":"1210.8252","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2012-10-31T07:37:32Z","cross_cats_sorted":[],"title_canon_sha256":"cfa0007e1dbaddda64dc7e58c60c88821f18220ca3469cc73c845360ad11f6b0","abstract_canon_sha256":"c92b9ce9b0beabc2586104c16119d1a191c6fff865012bd190d31b8e696a0126"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:37:21.939861Z","signature_b64":"ztQEVcUWpQA6akdeXLgBDcVJ0+M0ljfLK1807lpX1y2VcWVzWbpIXCYK+362hMkvRAMmViw8RmhP2n6+86bbBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6c78ce23aaa84c39bca9c68b85f05dca0988533237742bf242d4727fc0ad8d05","last_reissued_at":"2026-05-18T01:37:21.939171Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:37:21.939171Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Homotopy pullback of $A_n$-spaces and its applications to $A_n$-types of gauge groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Mitsunobu Tsutaya","submitted_at":"2012-10-31T07:37:32Z","abstract_excerpt":"We construct the homotopy pullback of $A_n$-spaces and show some universal property of it. As the first application, we review the Zabrodsky's result which states that for each prime $p$, there is a finite CW complex which admits an $A_{p-1}$-form but no $A_p$-form. As the second application, we investigate $A_n$-types of gauge groups. In particular, we give a new result on $A_n$-types of the gauge groups of principal $\\mathrm{SU}(2)$-bundles over $S^4$, which is a complete classification when they are localized away from 2."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.8252","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"1210.8252","created_at":"2026-05-18T01:37:21.939278+00:00"},{"alias_kind":"arxiv_version","alias_value":"1210.8252v2","created_at":"2026-05-18T01:37:21.939278+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.8252","created_at":"2026-05-18T01:37:21.939278+00:00"},{"alias_kind":"pith_short_12","alias_value":"NR4M4I5KVBGD","created_at":"2026-05-18T12:27:16.716162+00:00"},{"alias_kind":"pith_short_16","alias_value":"NR4M4I5KVBGDTPFJ","created_at":"2026-05-18T12:27:16.716162+00:00"},{"alias_kind":"pith_short_8","alias_value":"NR4M4I5K","created_at":"2026-05-18T12:27:16.716162+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/NR4M4I5KVBGDTPFJY2FYL4C5ZI","json":"https://pith.science/pith/NR4M4I5KVBGDTPFJY2FYL4C5ZI.json","graph_json":"https://pith.science/api/pith-number/NR4M4I5KVBGDTPFJY2FYL4C5ZI/graph.json","events_json":"https://pith.science/api/pith-number/NR4M4I5KVBGDTPFJY2FYL4C5ZI/events.json","paper":"https://pith.science/paper/NR4M4I5K"},"agent_actions":{"view_html":"https://pith.science/pith/NR4M4I5KVBGDTPFJY2FYL4C5ZI","download_json":"https://pith.science/pith/NR4M4I5KVBGDTPFJY2FYL4C5ZI.json","view_paper":"https://pith.science/paper/NR4M4I5K","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1210.8252&json=true","fetch_graph":"https://pith.science/api/pith-number/NR4M4I5KVBGDTPFJY2FYL4C5ZI/graph.json","fetch_events":"https://pith.science/api/pith-number/NR4M4I5KVBGDTPFJY2FYL4C5ZI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NR4M4I5KVBGDTPFJY2FYL4C5ZI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NR4M4I5KVBGDTPFJY2FYL4C5ZI/action/storage_attestation","attest_author":"https://pith.science/pith/NR4M4I5KVBGDTPFJY2FYL4C5ZI/action/author_attestation","sign_citation":"https://pith.science/pith/NR4M4I5KVBGDTPFJY2FYL4C5ZI/action/citation_signature","submit_replication":"https://pith.science/pith/NR4M4I5KVBGDTPFJY2FYL4C5ZI/action/replication_record"}},"created_at":"2026-05-18T01:37:21.939278+00:00","updated_at":"2026-05-18T01:37:21.939278+00:00"}