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Duval","submitted_at":"1998-09-30T20:09:09Z","abstract_excerpt":"\\newcommand{\\rhomi}[1]{\\widetilde{H}_{#1}} \\newcommand{\\rbeti}[1]{\\beta_{#1}} \\newcommand{\\kk}{\\mathbf k} \\newcommand{\\dimk}{\\dim_{\\kk}}\n  We show that algebraically shifting a pair of simplicial complexes weakly increases their relative homology Betti numbers in every dimension.\n  More precisely, let $\\Delta(K)$ denote the algebraically shifted complex of simplicial complex $K$, and let $\\rbeti{j}(K,L)=\\dimk \\rhomi{j}(K,L;\\kk)$ be the dimension of the $j$th reduced relative homology group over a field $\\kk$ of a pair of simplicial complexes $L \\subseteq K$. Then $\\rbeti{j}(K,L) \\leq \\rbeti{j}"},"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/9809195","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AT","submitted_at":"1998-09-30T20:09:09Z","cross_cats_sorted":[],"title_canon_sha256":"944aa2fe1a47c1ef237d5fd6b4d6eb59e1a6dfcde266f8ef38fe3cc79aaabd3b","abstract_canon_sha256":"b4c77768b355e07af46bdc98a418f887089c0280b60bdc1c0c19aae742499fdc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:33.729641Z","signature_b64":"5mfdBr2oLZHluCFUPJa6h+o1GsVXjFqHr9+8KDjhypQ5QkP7JpMmHPe5cZr38Kids8MrjlFzY6wHiUcqmHs8Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"107bb9243a56386ca357345b264f5013880985356fb91789de1cdf54b78e6248","last_reissued_at":"2026-05-18T01:05:33.728902Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:33.728902Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Algebraic Shifting Increases Relative Homology","license":"","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Art M. Duval","submitted_at":"1998-09-30T20:09:09Z","abstract_excerpt":"\\newcommand{\\rhomi}[1]{\\widetilde{H}_{#1}} \\newcommand{\\rbeti}[1]{\\beta_{#1}} \\newcommand{\\kk}{\\mathbf k} \\newcommand{\\dimk}{\\dim_{\\kk}}\n  We show that algebraically shifting a pair of simplicial complexes weakly increases their relative homology Betti numbers in every dimension.\n  More precisely, let $\\Delta(K)$ denote the algebraically shifted complex of simplicial complex $K$, and let $\\rbeti{j}(K,L)=\\dimk \\rhomi{j}(K,L;\\kk)$ be the dimension of the $j$th reduced relative homology group over a field $\\kk$ of a pair of simplicial complexes $L \\subseteq K$. 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