Configuration space (mathematics)
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.
is the set of n-tuples of pairwise distinct points in
: This space is generally endowed with the subspace topology from the inclusion of
[2] There is a natural action of the symmetric group
The intuition is that this action "forgets the names of the points".
Then the nth configuration space of X is denoted simply
[3] The n-strand braid group on a connected topological space X is the fundamental group of the nth unordered configuration space of X.
While the above definition is not the one that Emil Artin gave, Adolf Hurwitz implicitly defined the Artin braid groups as fundamental groups of configuration spaces of the complex plane considerably before Artin's definition (in 1891).
is a classifying space for the Artin braid group, and
is a manifold, its ordered configuration spaces are open subspaces of the powers of
The configuration space of distinct unordered points is also a manifold, while the configuration space of not necessarily distinct[clarification needed] unordered points is instead an orbifold.
It used to be an open question whether there were examples of compact manifolds which were homotopy equivalent but had non-homotopy equivalent configuration spaces: such an example was found only in 2005 by Riccardo Longoni and Paolo Salvatore.
That these configuration spaces are not homotopy equivalent was detected by Massey products in their respective universal covers.
[7] Homotopy invariance for configuration spaces of simply connected closed manifolds remains open in general, and has been proved to hold over the base field
[8][9] Real homotopy invariance of simply connected compact manifolds with simply connected boundary of dimension at least 4 was also proved.
This problem can be related to robotics and motion planning: one can imagine placing several robots on tracks and trying to navigate them to different positions without collision.
[11] and strong deformation retracts to a CW complex of dimension
deformation retract to non-positively curved cubical complexes of dimension at most
[13][14] One also defines the configuration space of a mechanical linkage with the graph
Such a graph is commonly assumed to be constructed as concatenation of rigid rods and hinges.
The configuration space of such a linkage is defined as the totality of all its admissible positions in the Euclidean space equipped with a proper metric.
The configuration space of a generic linkage is a smooth manifold, for example, for the trivial planar linkage made of
rigid rods connected with revolute joints, the configuration space is the n-torus
[15][16] The simplest singularity point in such configuration spaces is a product of a cone on a homogeneous quadratic hypersurface by a Euclidean space.
Such a singularity point emerges for linkages which can be divided into two sub-linkages such that their respective endpoints trace-paths intersect in a non-transverse manner, for example linkage which can be aligned (i.e. completely be folded into a line).
Many geometric applications require compact spaces, so one would like to compactify
, i.e., embed it as an open subset of a compact space with suitable properties.
Approaches to this problem have been given by Raoul Bott and Clifford Taubes,[18] as well as William Fulton and Robert MacPherson.
