Symmetric product (${\mathrm{Sym}}^g(X)$)

The $g$th symmetric product of a space $X$ is ${\mathrm{Sym}}^g(X)=X^g/S_g$, 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.