{"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 $L \\subseteq K$. Then $\\rbeti{j}(K,L) \\leq \\rbeti{j}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9809195","kind":"arxiv","version":1},"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"}