Algebraic & Geometric Topology 2 (2002) 1119–1145

DOI: 10.2140/agt.2002.2.1119

arXiv: math.GT/0209077

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=S¹ we show that: (1) exp_k(S¹) has the homotopy type of an odd dimensional sphere of dimension k or k-1; (2) the natural inclusion of exp_2k-1(S¹)≃ S^2k-1 into exp_2k(S¹)≃ S^2k-1 is multiplication by two on homology; (3) the complement exp_k(S¹)\exp_k-2(S¹) of the codimension two strata in exp_k(S¹) 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(S¹)→exp_k(S¹) is (deg f)^{⌊(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

Primary: 54B20

Secondary: 55Q52, 57M25

References

Forward citations

Publication

Received: 22 October 2002
Accepted: 30 November 2002
Published: 7 December 2002

Authors