# Spaces of Finite Subsets - homeomorphism type

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.

-

Yes, the manifold question is completely answered here. 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...
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 $S^1.$"
If you are ONLY interested in $S^1$ your comment is correct. Their result is for ANY closed manifold. – Igor Rivin Dec 29 '11 at 14:12
But this is part of the 'starting point' of my question, because for $S^1$ the result is that $\exp_nS^1$ is a closed manifold iff $n=1,3$. For other $n$ it could still be some other nice-enough space (like $\exp_2S^1$). – Chris Gerig Dec 29 '11 at 14:12
(sorry edited my comment right after you responded to it)... Yes for this followup question I am only interested in $S^1$. – Chris Gerig Dec 29 '11 at 14:13