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The spaces $exp_n(S^1)$, \exp_n(S^1)$, as well as the embeddings$exp_n(S^1) \exp_n(S^1) \subset \exp_{n+2}(S^1)$were studied by Christopher Tuffley in Finite subset spaces of$S^1$, Algebr. Geom. Topol. 2 (2002), 1119–1145, http://dx.doi.org/10.2140/agt.2002.2.1119; MR1998017 (2004f:54008), and, more recently, by Sadok Kallel and Denis Sjerve in Remarks on finite subset spaces, Homology, Homotopy Appl. 11 (2009), no. 2, 229–-250, http://www.intlpress.com/hha/v11/n2/a12/; MR2591920 (2011a:55019). In particular, based on an argument from Clifford H. Wagner's thesis (Symmetric, cyclic, and permutation products of manifolds, Dissertationes Math. (Rozprawy Mat.) 182 (1980); MR0605369 (82h:55021)), Kallel and Sjerve show that$exp_n(S^1)$\exp_n(S^1)$ is a closed manifold if and only if $n=1$ or $n=3$. Furthermore, Tuffley shows that $$\pi_1(\exp_{n+2}(S^1) \setminus \exp_{n}(S^1)) = \langle x, y \mid x^{n+2} = y^{n+1} \rangle.$$
The spaces $exp_n(S^1)$, as well as the embeddings $exp_n(S^1) \subset exp_{n+2}(S^1)$ were studied by Christopher Tuffley in Finite subset spaces of $S^1$, Algebr. Geom. Topol. 2 (2002), 1119–1145, http://dx.doi.org/10.2140/agt.2002.2.1119; MR1998017 (2004f:54008), and, more recently, by Sadok Kallel and Denis Sjerve in Remarks on finite subset spaces, Homology, Homotopy Appl. 11 (2009), no. 2, 229–-250, http://www.intlpress.com/hha/v11/n2/a12/; MR2591920 (2011a:55019). In particular, based on an argument from Clifford H. Wagner's thesis (Symmetric, cyclic, and permutation products of manifolds, Dissertationes Math. (Rozprawy Mat.) 182 (1980); MR0605369 (82h:55021)), Kallel and Sjerve show that $exp_n(S^1)$ is a closed manifold if and only if $n=1$ or $n=3$.