{"paper":{"title":"On diagonal digraphs, Koszul algebras and triangulations of homology spheres","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":["math.AT","math.CO","math.RT"],"primary_cat":"math.KT","authors_text":"Lev Mukoseev, Sergei O. Ivanov","submitted_at":"2024-05-08T01:29:57Z","abstract_excerpt":"The article is devoted to the magnitude homology of digraphs, with a primary focus on diagonal digraphs, i.e., digraphs whose magnitude homology is concentrated on the diagonal. For any digraph $G$, we provide a complete description of the second magnitude homology ${\\rm MH}_{2,k}(G)$. This allows us to define a combinatorial condition, denoted by $(\\mathcal{V}_\\ell)$, which is equivalent to the vanishing of ${\\rm MH}_{2,k}(G, \\mathbb{Z})$ for all $k > \\ell$. In particular, diagonal digraphs satisfy $(\\mathcal{V}_2)$. As a corollary, we obtain that the 2-dimensional CW-complex obtained from a "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2405.04748","kind":"arxiv","version":4},"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/2405.04748/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"}