{"paper":{"title":"A constructive proof for the simple connectedness of finite subset spaces","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.GN"],"primary_cat":"math.AT","authors_text":"J\\=anis Lazovskis","submitted_at":"2026-02-10T14:20:54Z","abstract_excerpt":"The space of all finite non-empty subsets of a topological space $X$, also known as the Ran space of $X$, is weakly contractible for $X$ path connected. We consider subspaces $\\mathrm{Ran}_{\\leqslant n}(X)$ of the Ran space given by all subsets of $X$ of size at most $n$, and their first homotopy groups. These groups are known to be trivial for $n\\geqslant 3$ when $X$ is a path connected CW-complex, though the proofs are not constructive. We show that the induced map $\\pi_1(\\mathrm{Ran}_{\\leqslant n}(X)) \\to \\pi_1(\\mathrm{Ran}_{\\leqslant n+2}(X))$ is trivial for all positive integers $n$, by e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2602.09815","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/2602.09815/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"}