{"paper":{"title":"An eigenvalue interlacing approach to Garland's method","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alan Lew","submitted_at":"2025-08-24T10:00:57Z","abstract_excerpt":"Let $X$ be a pure $d$-dimensional simplicial complex. For $0\\le k\\le d$, let $X(k)$ be the set of $k$-dimensional faces of $X$, let $\\tilde{L}_k(X)$ be the $k$-dimensional weighted total Laplacian operator on $X$, and let $\\tilde{H}_k(X;\\mathbb{R})$ be its $k$-dimensional reduced homology group with real coefficients. For $\\sigma\\in X$, let $\\text{lk}(X,\\sigma)$ be the link of $\\sigma$ in $X$. For a matrix $M$, we denote by $\\text{Spec}(M)$ the multi-set containing all the eigenvalues of $M$. We show that, for every $0\\le \\ell<k \\le d$, \\[\n  \\text{dim}(\\tilde{H}_k(X;\\mathbb{R}))\\le \\sum_{\\eta\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.17279","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2508.17279/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}