Finite subset spaces of $S^1$

Abstract

Given a topological space $X$ denote by ${exp}_k\left(X\right)$ the space of non-empty subsets of $X$ of size at most $k$, topologised as a quotient of $X^k$. This space may be regarded as a union over $1\le l\le k$ of configuration spaces of $l$ distinct unordered points in $X$. In the special case $X=S^1$ we show that: (1) ${exp}_k\left(S^1\right)$ has the homotopy type of an odd dimensional sphere of dimension $k$ or $k-1$; (2) the natural inclusion of ${exp}_{2k-1}\left(S^1\right)\simeq S^{2k-1}$ into ${exp}_{2k}\left(S^1\right)\simeq S^{2k-1}$ is multiplication by two on homology; (3) the complement ${exp}_k\left(S^1\right)\setminus {exp}_{k-2}\left(S^1\right)$ of the codimension two strata in ${exp}_k\left(S^1\right)$ has the homotopy type of a $\left(k-1,k\right)$–torus knot complement; and (4) the degree of an induced map ${exp}_k\left(f\right):{exp}_k\left(S^1\right)\to {exp}_k\left(S^1\right)$ is ${\left(degf\right)}^{\lfloor \left(k+1\right)∕2\rfloor }$ for $f:S^1\to S^1$. The first three results generalise known facts that ${exp}_2\left(S^1\right)$ is a Möbius strip with boundary ${exp}_1\left(S^1\right)$, and that ${exp}_3\left(S^1\right)$ is the three-sphere with ${exp}_1\left(S^1\right)$ inside it forming a trefoil knot.