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Let $\\{K_{i}{\\to} F_{i-1}{\\hookrightarrow} F_{i} \\,|\\, 1{\\le} i {\\le} n,\\, F_{0}{=} \\{\\ast\\} \\; F_{1}{=} \\Sigma{K_{1}} \\; \\text{and}\\; F_{n}{\\simeq} G \\}$ be a cone-decomposition of $G$ of length $m$ and $F'_{1}=\\Sigma{K'_{1}} \\subset F_{1}$ with $K'_{1} \\subset K_{1}$ which satisfy $F_{i}F'_{1} \\subset F_{i+1}$ up to homotopy for any $i$. Our main result is as follows: we have $\\operatorname{cat}(X) \\le m{+}1$, if firstl"},"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":"0712.3637","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2007-12-21T07:56:11Z","cross_cats_sorted":[],"title_canon_sha256":"f342681e87d8388ab1df15964469164cf009f464fdf9b65a8b452369d1e52fe2","abstract_canon_sha256":"121991f616136c72d5903b4f0c3b40724d5ef9fe0e14e9df3e17144480f3a74e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:24:57.684456Z","signature_b64":"uxi6bogBfpCtaw8d9EvdsjyQa4nE0xIGWc+a3hm6XSXs/uHZU/sK3RSkzuNeV21JPMtYf2mVc4Bo2FFZfRiqCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3444b6799a6613ee5bdf4115eb86794d77eb230a0b37dd4a1c613add40740407","last_reissued_at":"2026-05-18T02:24:57.683800Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:24:57.683800Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Lusternik-Schnirelmann category of SO(10)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Kai Kikuchi, Norio Iwase, Toshiyuki Miyauchi","submitted_at":"2007-12-21T07:56:11Z","abstract_excerpt":"Let $G$ be a compact connected Lie group and $p : E\\to \\Sigma A$ be a principal G-bundle with a characteristic map $\\alpha : A\\to G$, where $A=\\Sigma A_{0}$ for some $A_{0}$. Let $\\{K_{i}{\\to} F_{i-1}{\\hookrightarrow} F_{i} \\,|\\, 1{\\le} i {\\le} n,\\, F_{0}{=} \\{\\ast\\} \\; F_{1}{=} \\Sigma{K_{1}} \\; \\text{and}\\; F_{n}{\\simeq} G \\}$ be a cone-decomposition of $G$ of length $m$ and $F'_{1}=\\Sigma{K'_{1}} \\subset F_{1}$ with $K'_{1} \\subset K_{1}$ which satisfy $F_{i}F'_{1} \\subset F_{i+1}$ up to homotopy for any $i$. Our main result is as follows: we have $\\operatorname{cat}(X) \\le m{+}1$, if firstl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0712.3637","kind":"arxiv","version":3},"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":"0712.3637","created_at":"2026-05-18T02:24:57.683901+00:00"},{"alias_kind":"arxiv_version","alias_value":"0712.3637v3","created_at":"2026-05-18T02:24:57.683901+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0712.3637","created_at":"2026-05-18T02:24:57.683901+00:00"},{"alias_kind":"pith_short_12","alias_value":"GRCLM6M2MYJ6","created_at":"2026-05-18T12:25:55.427421+00:00"},{"alias_kind":"pith_short_16","alias_value":"GRCLM6M2MYJ64W67","created_at":"2026-05-18T12:25:55.427421+00:00"},{"alias_kind":"pith_short_8","alias_value":"GRCLM6M2","created_at":"2026-05-18T12:25:55.427421+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/GRCLM6M2MYJ64W67IEK6XBTZJV","json":"https://pith.science/pith/GRCLM6M2MYJ64W67IEK6XBTZJV.json","graph_json":"https://pith.science/api/pith-number/GRCLM6M2MYJ64W67IEK6XBTZJV/graph.json","events_json":"https://pith.science/api/pith-number/GRCLM6M2MYJ64W67IEK6XBTZJV/events.json","paper":"https://pith.science/paper/GRCLM6M2"},"agent_actions":{"view_html":"https://pith.science/pith/GRCLM6M2MYJ64W67IEK6XBTZJV","download_json":"https://pith.science/pith/GRCLM6M2MYJ64W67IEK6XBTZJV.json","view_paper":"https://pith.science/paper/GRCLM6M2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0712.3637&json=true","fetch_graph":"https://pith.science/api/pith-number/GRCLM6M2MYJ64W67IEK6XBTZJV/graph.json","fetch_events":"https://pith.science/api/pith-number/GRCLM6M2MYJ64W67IEK6XBTZJV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GRCLM6M2MYJ64W67IEK6XBTZJV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GRCLM6M2MYJ64W67IEK6XBTZJV/action/storage_attestation","attest_author":"https://pith.science/pith/GRCLM6M2MYJ64W67IEK6XBTZJV/action/author_attestation","sign_citation":"https://pith.science/pith/GRCLM6M2MYJ64W67IEK6XBTZJV/action/citation_signature","submit_replication":"https://pith.science/pith/GRCLM6M2MYJ64W67IEK6XBTZJV/action/replication_record"}},"created_at":"2026-05-18T02:24:57.683901+00:00","updated_at":"2026-05-18T02:24:57.683901+00:00"}