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braid theory
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 . For an elementary treatment along these lines, see the article on braid groups. Braid groups may also be given a deeper mathematical interpretation: as the fundamental group of certain configuration spaces.
== 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 ''X''''n'', 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 ''X''''n'' 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.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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