#### Volume 2, issue 2 (2002)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Author Index Editorial procedure Submission Guidelines Submission Page Author copyright form Subscriptions Contacts G&T Publications Author Index To Appear ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
Finite subset spaces of $S^1$

### Christopher Tuffley

Algebraic & Geometric Topology 2 (2002) 1119–1145
 arXiv: math.GT/0209077
##### 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 ${exp}_{k}\left(f\right):\phantom{\rule{0.3em}{0ex}}{exp}_{k}\left({S}^{1}\right)\to {exp}_{k}\left({S}^{1}\right)$ is ${\left(degf\right)}^{⌊\left(k+1\right)∕2⌋}$ for $f:\phantom{\rule{0.3em}{0ex}}{S}^{1}\to {S}^{1}$. 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.

##### Keywords
configuration spaces, finite subset spaces, symmetric product, circle
##### Mathematical Subject Classification 2000
Primary: 54B20
Secondary: 55Q52, 57M25