This is a followup to Spaces of Finite Subspaces. Just for convenience, $\exp_nX$ is the space whose underlying set is the set of nonempty subsets $S\subseteq X$ with $|S|\le n$.

As Alex Suciu pointed out in his answer to the previous post (referencing Chris Tuffley), $\exp_{2k-1}S^1$ and $\exp_{2k}S^1$ are homotopy equivalent to $S^{2k-1}$. And as for embeddings, $\exp_nS^1-\exp_{n-2}S^1$ has the homotopy type of an $(n-1,n)$-torus knot complement!

*Then can anything be said about homeomorphic properties?* Cliff Wagner showed in his thesis that $\exp_nS^1$ is a closed manifold iff $n=1$ or $n=3$, so generically they are not spheres. For instance, $\exp_2S^1$ is the Mobius band, yet this information is lost under homotopy; so it would be interesting to know what is lost in higher dimensions.