Configuration space (mathematics)

For a topological space ${\displaystyle X}$, the n^th (ordered) configuration space of X is the set of n-tuples of pairwise distinct points in ${\displaystyle X}$:

${\displaystyle \operatorname {Conf} _{n}(X):=\prod ^{n}X\smallsetminus \{(x_{1},x_{2},\ldots ,x_{n})\in X^{n}\mid x_{i}=x_{j}\ {\text{ for some }}i\neq j\}.}$[1]

This space is generally endowed with the subspace topology from the inclusion of ${\displaystyle \operatorname {Conf} _{n}(X)}$ into ${\displaystyle X^{n}}$. It is also sometimes denoted ${\displaystyle F(X,n)}$, ${\displaystyle F^{n}(X)}$, or ${\displaystyle {\mathcal {C}}^{n}(X)}$.[2]

There is a natural action of the symmetric group ${\displaystyle S_{n}}$ on the points in ${\displaystyle \operatorname {Conf} _{n}(X)}$ given by

${\displaystyle {\begin{aligned}S_{n}\times \operatorname {Conf} _{n}(X)&\longrightarrow \operatorname {Conf} _{n}(X)\\(\sigma ,x)&\longmapsto \sigma (x)=(x_{\sigma (1)},x_{\sigma (2)},\ldots ,x_{\sigma (n)}).\end{aligned}}}$

This action gives rise to the n^th unordered configuration space of X,

${\displaystyle \operatorname {UConf} _{n}(X):=\operatorname {Conf} _{n}(X)/S_{n},}$

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 ${\displaystyle {\mathcal {UC}}^{n}(X)}$.[2] The collection of unordered configuration spaces over all ${\displaystyle n}$ is the Ran space, and comes with a natural topology.

• The space of ordered configuration of two points in ${\displaystyle \mathbf {R} ^{2}}$ is homeomorphic to the product of the Euclidean 3-space with a circle, i.e. ${\displaystyle \operatorname {Conf} _{2}(\mathbf {R} ^{2})\cong \mathbf {R} ^{3}\times S^{1}}$.[2]
• More generally, the configuration space of two points in ${\displaystyle \mathbf {R} ^{n}}$ is homotopy equivalent to the sphere ${\displaystyle S^{n-1}}$. [4]
• The configuration space of ${\displaystyle n}$ points in ${\displaystyle \mathbf {R} ^{2}}$ is the classifying space of the ${\displaystyle n}$th braid group (see below).

The n-strand braid group on a connected topological space X is

${\displaystyle B_{n}(X):=\pi _{1}(\operatorname {UConf} _{n}(X)),}$

the fundamental group of the n^th unordered configuration space of X. The n-strand pure braid group on X is[2]

${\displaystyle P_{n}(X):=\pi _{1}(\operatorname {Conf} _{n}(X)).}$

The first studied braid groups were the Artin braid groups ${\displaystyle B_{n}\cong \pi _{1}(\operatorname {UConf} _{n}(\mathbf {R} ^{2}))}$. 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).[5]

It follows from this definition and the fact that ${\displaystyle \operatorname {Conf} _{n}(\mathbf {R} ^{2})}$ and ${\displaystyle \operatorname {UConf} _{n}(\mathbf {R} ^{2})}$ are Eilenberg–MacLane spaces of type ${\displaystyle K(\pi ,1)}$, that the unordered configuration space of the plane ${\displaystyle \operatorname {UConf} _{n}(\mathbf {R} ^{2})}$ is a classifying space for the Artin braid group, and ${\displaystyle \operatorname {Conf} _{n}(\mathbf {R} ^{2})}$ is a classifying space for the pure Artin braid group, when both are considered as discrete groups.[6]

If the original space ${\displaystyle X}$ is a manifold, its ordered configuration spaces are open subspaces of the powers of ${\displaystyle X}$ 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 ${\displaystyle \pi \colon E_{n}\to C_{n}}$ which is a sub-bundle of the trivial bundle ${\displaystyle C_{n}\times X\to C_{n}}$, and which has the property that the fiber over each point ${\displaystyle p\in C_{n}}$ is the n element subset of ${\displaystyle X}$ classified by p.

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.[11]

For any graph ${\displaystyle \Gamma }$, ${\displaystyle \operatorname {Conf} _{n}(\Gamma )}$ is an Eilenberg–MacLane space of type ${\displaystyle K(\pi ,1)}$[11] and strong deformation retracts to a CW complex of dimension ${\displaystyle b(\Gamma )}$, where ${\displaystyle b(\Gamma )}$ is the number of vertices of degree at least 3.[11][12] Moreover, ${\displaystyle \operatorname {UConf} _{n}(\Gamma )}$ and ${\displaystyle \operatorname {Conf} _{n}(\Gamma )}$ deformation retract to non-positively curved cubical complexes of dimension at most ${\displaystyle \min(n,b(\Gamma ))}$.[13][14]

• Configuration space (physics)
• State space (physics)