A constructive proof for the simple connectedness of finite subset spaces

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 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 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 $\pi_1(\mathrm{Ran}_{\leqslant n}(X))$, for all $n\geqslant 4$.