Finite subset spaces of S1

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 $1 \leq l \leq k$ 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 ${exp}_{2 k - 1} (S^{1}) ≃ S^{2 k - 1}$ into ${exp}_{2 k} (S^{1}) ≃ S^{2 k - 1}$ 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 ${exp}_{k} (f) : {exp}_{k} (S^{1}) \to {exp}_{k} (S^{1})$ is ${(deg f)}^{⌊ (k + 1) ∕ 2 ⌋}$ for $f : S^{1} \to S^{1}$ . The first three results generalise known facts that ${exp}_{2} (S^{1})$ is a Möbius 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

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

Volume 2, issue 2 (2002)

Christopher Tuffley