Algebraic and Geometric Topology 2 (2002), paper no. 43, pages 1119-1145.

Finite subset spaces of S^1

Christopher Tuffley

Abstract. Given a topological space X denote by exp_k(X) 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 0 < l < k+1 of configuration spaces of l distinct unordered points in X. In the special case X=S^1 we show that: (1) exp_k(S^1) has the homotopy type of an odd dimensional sphere of dimension k or k-1; (2) the natural inclusion of exp_{2k-1}(S^1) h.e. S^{2k-1} into exp_2k(S^1) h.e. S^{2k-1} is multiplication by two on homology; (3) the complement exp_k(S^1)-exp_{k-2}(S^1) of the codimension two strata in exp_k(S^1) has the homotopy type of a (k-1,k)-torus knot complement; and (4) the degree of an induced map exp_k(f): exp_k(S^1)-->exp_k(S^1) is (deg f)^[(k+1)/2] for f: S^1-->S^1. The first three results generalise known facts that exp_2(S^1) is a Moebius strip with boundary exp_1(S^1), and that exp_3(S^1) is the three-sphere with exp_1(S^1) inside it forming a trefoil knot.

Keywords. Configuration spaces, finite subset spaces, symmetric product, circle

AMS subject classification. Primary: 54B20. Secondary: 55Q52, 57M25.

DOI: 10.2140/agt.2002.2.1119

E-print: arXiv:math.GT/0209077

Submitted: 22 October 2002. Accepted: 30 November 2002. Published: 7 December 2002.

Notes on file formats

Christopher Tuffley
Department of Mathematics, University of California
Berkeley, CA 94720, U.S.A.

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