Braid Theory - Braids As Fundamental Groups

Braids As Fundamental Groups

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

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

With this definition, then, we can call the braid group of X with n strings the fundamental group of Y (for any choice of base point – this is well-defined up to isomorphism). The case where X is the Euclidean plane is the original one of Artin. In some cases it can be shown that the higher homotopy groups of Y are trivial.