Configuration Space - Configuration Spaces in Mathematics

Configuration Spaces in Mathematics

In mathematics a configuration space refers to a broad family of constructions closely related to the state space notion in physics. The most common notion of configuration space in mathematics is the set of n-element subsets of a topological space . This set is given a topology by considering it as the quotient where and is the symmetric group acting by permuting the coordinates of . Typically, is called the configuration space of n unordered points in and is called the configuration space of n ordered or coloured points in ; the space of n ordered not necessarily distinct points is simply

If the original space is a manifold, the configuration space of distinct, unordered points is also a manifold, while the configuration space of not necessarily distinct unordered points is instead an orbifold.

Configuration spaces are related to braid theory, where the braid group is considered as the fundamental group of the space .

A configuration space is a type of classifying space or (fine) moduli space. In particular, there is a universal bundle which is a subbundle of the trivial bundle, and which has the property that the fiber over each point is the n element subset of classified by p.

The homotopy type of configuration spaces is not homotopy invariant – for example, that the spaces are not homotopic for any two distinct values of . For instance, is not connected, is a, and is simply connected for .

It used to be an open question whether there were examples of compact manifolds which were homotopic but had non-homotopic configuration spaces: such an example was found only in 2005 by Longini and Salvatore. Their example are two three-dimensional lens spaces, and the configuration spaces of at least two points in them. That these configuration spaces are not homotopic was detected by Massey products in their respective universal covers.