Constructions of contractibility in 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 ${\mathrm{Ran}}_{⩽n}(X)$ 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 $n⩾3$, though the proofs are not constructive. We draw an explicit contraction in ${\mathrm{Ran}}_{⩽3}(S^1)$ of the generator of ${\pi }_1(S^1)$, which we then use to conclude that the induced maps ${\pi }_1({\mathrm{Ran}}_{⩽n}(X))\to {\pi }_1({\mathrm{Ran}}_{⩽2n+1}(X))$ and ${\pi }_1({\mathrm{Ran}}_{⩽n}(X))\to {\pi }_1({\mathrm{Ran}}_{⩽n+2}(X))$ are trivial for all positive integers $n$. Pathological maps with infinitely many strata changes in ${\mathrm{Ran}}_{⩽n}(X)$ are resolved via locally contractible neighbourhoods and exit paths. From this we conclude the triviality of ${\pi }_1({\mathrm{Ran}}_{⩽n}(X))$, for all $n⩾3$.

arXiv commentMajor revision, simpler construction, stronger results. Comments welcome