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
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