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 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.