Symmetric products of the circle

Source abstract

The n th symmetric product of a topological space, X , is defined to be the quotient of the Cartesian product X n by the action of the symmetric group which permutes the factors. Even if X is a manifold, this product is, in general, not a manifold. The purpose of this note is to determine these products when X is the circle, S 1 , and to show that they are manifolds with boundary.