A path in the formula_4-fold symmetric product is the abstract way of discussing formula_4 points of formula_3, considered as an unordered formula_4-tuple, independently tracing out formula_4 strings. Since we must require that the strings never pass through each other, it is necessary that we pass to the subspace formula_16 of the symmetric product, of orbits of formula_4-tuples of "distinct" points. That is, we remove all the subspaces of formula_6 defined by conditions formula_19 for all formula_20. This is invariant under the symmetric group, and formula_16 is the quotient by the symmetric group of the non-excluded formula_4-tuples. Under the dimension condition formula_16 will be connected.