Configuration space (mathematics)

In mathematics, a configuration space is a construction closely related to state spaces or phase spaces in physics. In physics, these are used to describe the state of a whole system as a single point in a high-dimensional space. In mathematics, they are used to describe assignments of a collection of points to positions in a topological space. More specifically, configuration spaces in mathematics are particular examples of configuration spaces in physics in the particular case of several non-colliding particles. For a topological space , the nth (ordered) configuration space of X is the set of n-tuples of pairwise distinct points in : This space is generally endowed with the subspace topology from the inclusion of into . It is also sometimes denoted , , or . There is a natural action of the symmetric group on the points in given by This action gives rise to the th unordered configuration space of , which is the orbit space of that action. The intuition is that this action "forgets the names of the points". The unordered configuration space is sometimes denoted , , or . The collection of unordered configuration spaces over all is the Ran space, and comes with a natural topology. For a topological space and a finite set , the configuration space of with particles labeled by is For , define . Then the th configuration space of X is , and is denoted simply . The space of ordered configuration of two points in is homeomorphic to the product of the Euclidean 3-space with a circle, i.e. . More generally, the configuration space of two points in is homotopy equivalent to the sphere . The configuration space of points in is the classifying space of the th braid group (see below). Braid group The -strand braid group on a connected topological space is the fundamental group of the th unordered configuration space of . The -strand pure braid group on is The first studied braid groups were the Artin braid groups .