$\exp\_nX$ is the space whose underlying set is the set of nonempty subsets $S\subseteq X$ with $|S|\le n$. Its topology is the quotient one inherited from the map $X^{\oplus n}\rightarrow\exp\_nX$ given by $(x\_1,\ldots,x\_n)\mapsto\lbrace x\_1\rbrace\cup\cdots\lbrace x\_n\rbrace$. And $\exp\_{m\le n}X$ is canonically embedded in it.


Interestingly, for the case $X=S^1$, we have $\exp\_2S^1\approx M\ddot{o}$ (homeomorphic mobius band). Etienne Ghys saw this by considering the mobius band as $\mathbb{R}P^2$ minus the open disk (with $S^1$ as the disk's boundary) and mapping $p\in M\ddot{o}$ to the set of tangency points of the lines tangent to $S^1$ and intersecting $p$. And from this we see that $\exp\_1S^1$ is the boundary of the band and not the meridian circle. Now Raul Bott showed that $\exp\_3S^1\approx S^3$ (*On the Third Symmetric Potency of $S^1$*), and someone else showed that $\exp\_1S^1\subset S^3$ is the trefoil knot. Furthermore, $\exp\_2S^1$ is a Seifert surface of $\exp\_1S^1\subset S^3$.


My two curious questions: *What happens for $n\ge 3$ and the corresponding embeddings? Are there interesting results for other $X$?*



Spaces of Finite Subsets
The spaces $\exp\_n(S^1)$, as well as the embeddings $\exp\_n(S^1) \subset \exp\_{n+2}(S^1)$ were studied by Christopher Tuffley in *Finite subset spaces of $S^1$,* Algebr. Geom. Topol. **2** (2002), 1119–1145, <http://dx.doi.org/10.2140/agt.2002.2.1119>; MR1998017 (2004f:54008), and, more recently, by Sadok Kallel and Denis Sjerve in *Remarks on finite subset spaces*, Homology, Homotopy Appl. **11** (2009), no. 2, 229–-250, <http://www.intlpress.com/hha/v11/n2/a12/>; MR2591920 (2011a:55019). 


In particular, based on an argument from Clifford H. Wagner's thesis (*Symmetric, cyclic, and permutation products of manifolds*, Dissertationes Math. (Rozprawy Mat.) **182** (1980); MR0605369 (82h:55021)), Kallel and Sjerve show that $\exp\_n(S^1)$ is a closed manifold if and only if $n=1$ or $n=3$. Furthermore, Tuffley shows that
$$
\pi\_1(\exp\_{n+2}(S^1) \setminus \exp\_{n}(S^1)) = \langle x, y \mid x^{n+2} = y^{n+1} \rangle.
$$


