To explain how to reduce a braid group in the sense of Artin to a fundamental group, we consider a connected manifold formula_3 of dimension at least 2. The "symmetric product" of formula_4 copies of formula_3 means the quotient of formula_6, the formula_4-fold Cartesian product of formula_3 by the permutation action of the symmetric group on formula_4 strands operating on the indices of coordinates. That is, an ordered formula_4-tuple is in the same orbit as any other that is a re-ordered version of it.