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Finite subset spaces of S¹

Christopher Tuffley

Algebraic & Geometric Topology 2 (2002) 1119–1145

DOI: 10.2140/agt.2002.2.1119

arXiv: math.GT/0209077


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 1≤ l ≤ k 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 k-1; (2) the natural inclusion of exp2k-1(S1)≃ S2k-1 into exp2k(S1)≃ S2k-1 is multiplication by two on homology; (3) the complement expk(S1)\expk-2(S1) of the codimension two strata in expk(S1) has the homotopy type of a (k-1,k)–torus knot complement; and (4) the degree of an induced map expk(f): expk(S1)→expk(S1) is (deg f)⌊(k+1)/2⌋ for f: S1→ S1. 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.


configuration spaces, finite subset spaces, symmetric product, circle

Mathematical Subject Classification

Primary: 54B20

Secondary: 55Q52, 57M25

Forward citations

Received: 22 October 2002
Accepted: 30 November 2002
Published: 7 December 2002

Christopher Tuffley
Department of Mathematics
University of California
Berkeley, CA 94720