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.
Definition.
For a topological space formula_1, the "n"th (ordered) configuration space of X is the set of "n"-tuples of pairwise distinct points in formula_1:
This space is generally endowed with the subspace topology from the inclusion of formula_4 into formula_5. It is also sometimes denoted formula_6, formula_7, or formula_8.
There is a natural action of the symmetric group formula_9 on the points in formula_4 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 formula_13, formula_14, or formula_15. The collection of unordered configuration spaces over all formula_16 is the Ran space, and comes with a natural topology.
Alternative formulations.
For a topological space formula_1 and a finite set formula_18, the configuration space of with particles labeled by is
For formula_20, define formula_21. Then the th configuration space of "X" is formula_22, and is denoted simply formula_4.
Connection to braid groups.
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 formula_33. 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).
It follows from this definition and the fact that formula_34 and formula_35 are Eilenberg–MacLane spaces of type formula_36, that the unordered configuration space of the plane formula_35 is a classifying space for the Artin braid group, and formula_34 is a classifying space for the pure Artin braid group, when both are considered as discrete groups.
Configuration spaces of manifolds.
If the original space formula_1 is a manifold, its ordered configuration spaces are open subspaces of the powers of formula_1 and are thus themselves manifolds. 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.
A configuration space is a type of classifying space or (fine) moduli space. In particular, there is a universal bundle formula_41 which is a sub-bundle of the trivial bundle formula_42, and which has the property that the fiber over each point formula_43 is the "n" element subset of formula_44 classified by "p".
Homotopy invariance.
The homotopy type of configuration spaces is not homotopy invariant. For example, the spaces formula_45 are not homotopy equivalent for any two distinct values of formula_46: formula_47 is empty for formula_48, formula_49 is not connected for formula_48, formula_51 is an Eilenberg–MacLane space of type formula_36, and formula_45 is simply connected for formula_54.
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. 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 homotopy equivalent was detected by Massey products in their respective universal covers. Homotopy invariance for configuration spaces of simply connected closed manifolds remains open in general, and has been proved to hold over the base field formula_55. Real homotopy invariance of simply connected compact manifolds with simply connected boundary of dimension at least 4 was also proved.
Configuration spaces of graphs.
Some results are particular to configuration spaces of graphs. 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. The tracks correspond to (the edges of) a graph, the robots correspond to particles, and successful navigation corresponds to a path in the configuration space of that graph.
For any graph formula_56, formula_57 is an Eilenberg–MacLane space of type formula_36 and strong deformation retracts to a CW complex of dimension formula_59, where formula_59 is the number of vertices of degree at least 3. Moreover, formula_61 and formula_57 deformation retract to non-positively curved cubical complexes of dimension at most formula_63.
Configuration spaces of mechanical linkages.
One also defines the configuration space of a mechanical linkage with the graph formula_56 its underlying geometry. 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 formula_16 rigid rods connected with revolute joints, the configuration space is the n-torus formula_66.
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).