Volume 2, issue 2 (2002)

Download this article
For printing
Recent Issues

Volume 20
Issue 7, 3219–3760
Issue 6, 2687–3218
Issue 5, 2145–2685
Issue 4, 1601–2143
Issue 3, 1073–1600
Issue 2, 531–1072
Issue 1, 1–529

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editorial Interests
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Author Index
To Appear
 
Other MSP Journals
Finite subset spaces of $S^1$

Christopher Tuffley

Algebraic & Geometric Topology 2 (2002) 1119–1145

arXiv: math.GT/0209077

Abstract

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 exp2k1(S1) S2k1 into exp2k(S1) S2k1 is multiplication by two on homology; (3) the complement expk(S1) expk2(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.

Keywords
configuration spaces, finite subset spaces, symmetric product, circle
Mathematical Subject Classification 2000
Primary: 54B20
Secondary: 55Q52, 57M25
References
Forward citations
Publication
Received: 22 October 2002
Accepted: 30 November 2002
Published: 7 December 2002
Authors
Christopher Tuffley
Department of Mathematics
University of California
Berkeley, CA 94720
USA