Simple connectedness of the Ran space

TL;DR

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Contribution

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Abstract

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 present results on their first homotopy groups. In particular, 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 $n$, and even more, show that $\pi_1(\mathrm{Ran}_{\leqslant n}(X)) = 0$ for all $n\geqslant 4$, by explicitly drawing the path homotopies that contract any loop to a point.