Volume 2, issue 2 (2002)

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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 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 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 (k 1,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 : 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.

Keywords
configuration spaces, finite subset spaces, symmetric product, circle
Mathematical Subject Classification 2000
Primary: 54B20
Secondary: 55Q52, 57M25
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Publication
Received: 22 October 2002
Accepted: 30 November 2002
Published: 7 December 2002
Authors
Christopher Tuffley
Department of Mathematics
University of California
Berkeley, CA 94720
USA