#### 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.

Email: tuffley@math.berkeley.edu

AGT home page

## Archival Version

**These pages are not updated anymore.
They reflect the state of
.
For the current production of this journal, please refer to
http://msp.warwick.ac.uk/.
**