{"paper":{"title":"Vertex Bounds in Triangulated $d$-Manifolds and an Application to 4-Manifold Complexity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GT","authors_text":"Jonathan Spreer, Lucy Tobin","submitted_at":"2024-01-20T07:17:04Z","abstract_excerpt":"We investigate face numbers of generalised triangulations of manifolds in arbitrary dimensions. This is motivated by the study of connections between the combinatorics of triangulations and topological properties of their underlying manifolds. For an $n$-facet triangulation of an odd-dimensional $d$-manifold with $n \\geq d$, we prove that the number of vertices satisfies $v \\leq n + \\frac{d - 1}{2}$. Moreover, we show that this bound is tight for all odd $d$ and all $n \\geq d$. For even dimensions, we conjecture the bound $v \\leq \\frac{n}{2} + d$. We prove that, if true, the bound is tight. Al"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2401.11152","kind":"arxiv","version":2},"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/2401.11152/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"}