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$\exp_nX$ 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 three two curious questions: What happens for $n\ge 3$ and the corresponding embeddings? Are there interesting results for other $X$?What if we consider differential structures?

[[Addendum]]: As Alex Suciu pointed out in his answer below (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 structures? 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.

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$\exp_nX$ 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 three curious questions: What happens for $n\ge 3$ and the corresponding embeddings? Are there interesting results for other $X$? What if we consider differential structures?

[[Addendum]]: As Alex Suciu pointed out in his answer below (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 structures? 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.

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$\exp_nX$ is the space whose underlying set is the set of nonempty subsets $S\subseteq X$ with $|S|\le n$. Its topology is the quottient 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 three curious questions: What happens for $n\ge 3$ and the corrseponding corresponding embeddings? Are there interesting results for other $X$? What if we consider differential structures?

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