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C(G)) as the complex associated to the poset of pairwise non-commuting (resp. commuting) sets in G.\n  We observe that NC(G) has only one positive dimensional connected component, which we call BNC(G), and we prove that BNC(G) is simply connected.\n  Our main result is a simplicial decomposition formula for BNC(G) which follows from a result of A. Bjorner, M. Wachs and V. Welker on inflated simplicial complexes. As a corollary, we obtain that if G has a nontrivi"},"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":"math/0005301","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AT","submitted_at":"2000-05-24T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"96d35441e354843aed8b36116424f0fb997b0b33f43e70b1fab1bea611b4b966","abstract_canon_sha256":"b87ae6846f2ad88572e29488b118455b7809610912534134ff3339594c86919d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:26:32.847987Z","signature_b64":"SL6ltGjQLKa8eJ266YYenrSKD2OgiJBbnp+bdrSC14LEP9ZHNJ8WmmFmd2s1X60jed6hfVmR6A3EwMjFs0+LAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"29bb0a4210f890a5b39ff967be802a5673581d908b8b73557c80009544a4e7d8","last_reissued_at":"2026-05-18T03:26:32.847645Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:26:32.847645Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On commuting and non-commuting complexes","license":"","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Erg\\\"un Yal\\c{c}in, Jonathan Pakianathan","submitted_at":"2000-05-24T00:00:00Z","abstract_excerpt":"In this paper we study various simplicial complexes associated to the commutative structure of a finite group G. 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