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...

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

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...

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 $S^1.$"

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...

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...

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