{"paper":{"title":"$A_\\infty$ persistent homology estimates the topology from pointcloud datasets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Anastasios Stefanou, Francisco Belch\\'i","submitted_at":"2019-02-25T08:35:52Z","abstract_excerpt":"Let $X$ be a closed subspace of a metric space $M$. Under mild hypotheses, one can estimate the Betti numbers of $X$ from a finite set $P \\subset M$ of points approximating $X$. In this paper, we show that one can also use $P$ to estimate much more detailed topological properties of $X$. These properties are computed via $A_\\infty$-structures, and are therefore related to the cup and Massey products of $X$, its loop space $\\Omega X$, its formality, linking numbers, etc.\n  Additionally, we study the following setting: given a continuous function $f \\colon Y \\longrightarrow \\mathbb R$ on a topol"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.09138","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"}