Configuration space (mathematics)

The configuration space of all unordered pairs of distinct points on the circle is the Möbius strip.

In mathematics a configuration space (also known as Fadell's configuration space) is any of several constructions closely related to the state space notion 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 $X$, the n^th (ordered) configuration space of X is the set of n-tuples of pairwise distinct points in $X$:

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

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

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

${\begin{aligned}S_{n}\times \operatorname {Conf} _{n}(X)&\longrightarrow \operatorname {Conf} _{n}(X)\\(\sigma ,x)&\longmapsto \sigma (x)\end{aligned}}$

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

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

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

Alternative formulations

For a topological space $X$ and a finite set $S$, the configuration space of X with particles labeled by S is

$\operatorname {Conf} _{S}(X):=\{f\mid f:S\hookrightarrow X{\text{ is injective}}\}$

For $n\in \mathbf {N}$, define $\mathbf {n} :=\{1,2,\ldots ,n\}$. Then the n^th configuration space of X is $\operatorname {Conf} _{\mathbf {n} }(X)$, and is denoted simply $\operatorname {Conf} _{n}(X)$.

Examples

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

Connection to braid groups

Main article: Braid group

The n-strand braid group on a connected topological space X is $B_{n}(X):=\pi _{1}(\operatorname {UConf} _{n}(X))$, the fundamental group of the n^th unordered configuration space of X. The n-strand pure braid group on X is $P_{n}(X):=\pi _{1}(\operatorname {Conf} _{n}(X))$.

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

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

Configuration spaces of manifolds

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.

A configuration space is a type of classifying space or (fine) moduli space. In particular, there is a universal bundle $\pi \colon E_{n}\to C_{n}$ which is a sub-bundle of the trivial bundle $C_{n}\times X\to C_{n}$, and which has the property that the fiber over each point $p\in C_{n}$ is the n element subset of $X$ classified by p.

The homotopy type of configuration spaces is not homotopy invariant – for example, the spaces $\operatorname {Conf} _{n}({\mathbb {R}}^{m})$ are not homotopy equivalent for any two distinct values of $m$. For instance, $\operatorname {Conf} _{n}({\mathbb {R}})$ is not connected, $\operatorname {Conf} _{n}({\mathbb {R}}^{2})$ is an Eilenberg–MacLane space of type $K(\pi ,1)$, and $\operatorname {Conf} _{n}({\mathbb {R}}^{m})$ is simply connected for $m\geq 3$.

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 Paolo Longoni and Riccardo 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 $\mathbf {R}$.

Configuration spaces of graphs

Some results are particular to configuration spaces of graphs. Robert Ghrist motivates these problems as 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 $\Gamma$,

• $\operatorname {Conf} _{n}(\Gamma )$ are Eilenberg–MacLane spaces of type $K(\pi ,1)$;
• $\operatorname {Conf} _{n}(\Gamma )$ strong deformation retracts to a subspace of dimension $b(\Gamma )$, where $b(\Gamma )$ is the number of vertices of degree at least 3;
• $\operatorname {UConf} _{n}(\Gamma )$ and $\operatorname {Conf} _{n}(\Gamma )$ deformation retract to non-positively curved cubical complexes of dimension at most $\min(n,b(\Gamma ))$.

See also

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