This is a followup to [Spaces of Finite Subspaces](https://mathoverflow.net/questions/84245/spaces-of-finite-subsets). 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.



Spaces of Finite Subsets - homeomorphism type
Yes, the manifold question is completely answered [here](http://arxiv.org/pdf/0902.3773v2). Namely $\exp\_n X^k$ (where $X^k$ is a $k$-manifold) is a manifold *if and only if* $k=1, n=3$ or $k=n=2.$ This is Theorem 1.3 in the referenced paper. **EDIT** Also, of course if $n=1,$ though the authors overlook this...


**EDIT** By the way, some very nice papers on the subject have been written by Chris Tuffley (a couple seem to be in AGT).


**ANOTHER EDIT** In particular, Tuffley gives the simplicial complex structure of $\exp\_n S^1$ explicitly, and also describes the "complement" of $\exp\_{n-2}S^1$ in $\exp\_n S^1,$ which is already interesting in the case $n=3$ (it's the trefoil knot complement). From Tuffley's thing you can, at least in principle, answer all homeomorpism-related questions (I am referring to the paper with the illogical title: "Finite Subset Spaces of $S^1.$"


