Closed braid

In topology, a branch of mathematics, braid theory is an abstract geometric theory studying the everyday braid concept, and some generalizations. The idea is that braids can be organized into groups, in which the group operation is 'do the first braid on a set of strings, and then follow it with a second on the twisted strings'. Such groups may be described by explicit presentations, as was shown by Emil Artin (1947). For an elementary treatment along these lines, see the article on braid groups. Braid groups are also understood by a deeper mathematical interpretation: as the fundamental group of certain configuration spaces.

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 by the permutation action of the symmetric group on n strands 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.

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