Finite subset spaces of S 1

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 1 ≤ l ≤ k 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 ≃ S 2k−1 into exp 2k S 1 ≃ 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 Möbius 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.