Given a topological space denote by the space of non-empty subsets of of size at most , topologised as a quotient of . This space may be regarded as a union over of configuration spaces of distinct unordered points in . In the special case we show that: (1) has the homotopy type of an odd dimensional sphere of dimension or ; (2) the natural inclusion of into is multiplication by two on homology; (3) the complement of the codimension two strata in has the homotopy type of a –torus knot complement; and (4) the degree of an induced map is for . The first three results generalise known facts that is a Möbius strip with boundary , and that is the three-sphere with inside it forming a trefoil knot.
"Finite subset spaces of $S^1$." Algebr. Geom. Topol. 2 (2) 1119 - 1145, 2002. https://doi.org/10.2140/agt.2002.2.1119