A constructive proof for the simple connectedness of finite subset spaces
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866910274878963712 |
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| author | Lazovskis, Jānis |
| author_facet | Lazovskis, Jānis |
| contents | 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 $π_1(\mathrm{Ran}_{\leqslant n}(X)) \to π_1(\mathrm{Ran}_{\leqslant n+2}(X))$ is trivial for all positive integers $n$, by explicitly drawing the path homotopies that contract any loop in $X$ to a point. From this we get a constructive proof for the triviality of $π_1(\mathrm{Ran}_{\leqslant n}(X))$, for all $n\geqslant 4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_09815 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A constructive proof for the simple connectedness of finite subset spaces Lazovskis, Jānis Algebraic Topology General Topology 55R80 (primary), 55P15, 57M05, 55Q52 (secondary) 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 $π_1(\mathrm{Ran}_{\leqslant n}(X)) \to π_1(\mathrm{Ran}_{\leqslant n+2}(X))$ is trivial for all positive integers $n$, by explicitly drawing the path homotopies that contract any loop in $X$ to a point. From this we get a constructive proof for the triviality of $π_1(\mathrm{Ran}_{\leqslant n}(X))$, for all $n\geqslant 4$. |
| title | A constructive proof for the simple connectedness of finite subset spaces |
| topic | Algebraic Topology General Topology 55R80 (primary), 55P15, 57M05, 55Q52 (secondary) |
| url | https://arxiv.org/abs/2602.09815 |