Open Access
Translator Disclaimer
2002 Finite subset spaces of $S^1$
Christopher Tuffley
Algebr. Geom. Topol. 2(2): 1119-1145 (2002). DOI: 10.2140/agt.2002.2.1119


Given a topological space X denote by expk(X) the space of non-empty subsets of X of size at most k, topologised as a quotient of Xk. This space may be regarded as a union over 1lk of configuration spaces of l distinct unordered points in X. In the special case X=S1 we show that: (1) expk(S1) has the homotopy type of an odd dimensional sphere of dimension k or k1; (2) the natural inclusion of exp2k1(S1)S2k1 into exp2k(S1)S2k1 is multiplication by two on homology; (3) the complement expk(S1) expk2(S1) of the codimension two strata in expk(S1) has the homotopy type of a (k1,k)–torus knot complement; and (4) the degree of an induced map expk(f):expk(S1) expk(S1) is (degf)(k+1)2 for f:S1S1. The first three results generalise known facts that exp2(S1) is a Möbius strip with boundary exp1(S1), and that exp3(S1) is the three-sphere with exp1(S1) inside it forming a trefoil knot.


Download Citation

Christopher Tuffley. "Finite subset spaces of $S^1$." Algebr. Geom. Topol. 2 (2) 1119 - 1145, 2002.


Received: 22 October 2002; Accepted: 30 November 2002; Published: 2002
First available in Project Euclid: 21 December 2017

zbMATH: 1015.55009
MathSciNet: MR1998017
Digital Object Identifier: 10.2140/agt.2002.2.1119

Primary: 54B20
Secondary: 55Q52, 57M25

Rights: Copyright © 2002 Mathematical Sciences Publishers


Vol.2 • No. 2 • 2002
Back to Top