Finite subset spaces of S1

Tuffley, Christopher¹

¹Department of Mathematics, University of California, Berkeley, CA 94720, USA

Algebraic and Geometric Topology, Tome 2 (2002) no. 2, pp. 1119-1145

Résumé

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 (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.

Détail

DOI : 10.2140/agt.2002.2.1119

Keywords: configuration spaces, finite subset spaces, symmetric product, circle

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