= Symmetric product
{title2=$\operatorname{Sym}^g(X)$}
The $g$th symmetric product of a space $X$ is $\operatorname{Sym}^g(X)=X^g/S_g$, where the <symmetric group> permutes the factors. Its points are unordered $g$-tuples, with repetitions allowed. If $X$ is a <Riemann surface>, elementary symmetric functions in local complex coordinates make its symmetric product a smooth complex manifold, including along the locus of repeated points. A collection of disjoint circles in $X$ gives a product torus in this symmetric product.
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