TITLE: Quotient of Three Dimensional Torus by Permutation on Coordinates
QUESTION [14 upvotes]: The Mobius Strip can be realized as a quotient of $T = (S^1)^2$ via the identifications $(x,y) \sim (y,x)$. 
I tried to generalized this concept to a higher dimension, and consider the quotient of $(S^1)^3$ by the action of the symmetric group $S_3$ on the coordinates. 
I was able to compute the homology of this space: $H_n = \mathbb{Z}$ for $n = 0,1$, and 0 otherwise (with reduced homology being 0 at $n=0$ as well).
Even with this information I wasn't able to identify said space in any other way. Is it well known, or, can it be described in any other fashion? What can be said about higher dimensions?

REPLY [12 votes]: What you are describing are the symmetric powers of $S^1$. The symmetric power $SP^n(S^1)$ is a fibre bundle over $S^1$ with fibre an $(n-1)$-simplex. (In particular your calculation of the homology for $n=3$ checks out.) This bundle is orientable if $n$ is odd, and non-orientable if $n$ is even.
This result is due to H. R. Morton, and is nicely written up at the nLab page https://ncatlab.org/nlab/show/symmetric+product+of+circles