View source for Configuration space (mathematics)

[[File:Moebius_Surface_1_Display_Small.png|thumb|The configuration space of all unordered pairs of points on the circle is the [[Möbius strip]].]]
<!-- Deleted image removed: [[File:Topological III by Robert R. Wilson, at Harvard University.JPG|upright|thumb|The configuration space of 3 not necessarily distinct points on the circle <math>T^3/S_3,</math> is the above orbifold.]] -->

In [[mathematics]], a '''configuration space''' is a construction closely related to [[state space]]s or [[phase space]]s 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 space (physics)|configuration spaces in physics]] in the particular case of several non-colliding particles.

== Definition ==
For a topological space <math>X</math> and a positive integer <math>n</math>, let <math>X^n</math> be the [[Cartesian product]] of <math>n</math> copies of <math>X</math>, equipped with the [[product topology]]. The '''''n''<sup>th</sup> (ordered) configuration space of <math>X</math>''' is the set of ''n''-[[tuple]]s of pairwise distinct points in <math>X</math>:

:<math>\operatorname{Conf}_n(X) := X^n \smallsetminus \{(x_1,x_2,\ldots,x_n)\in X^n \mid x_i=x_j\ \text{for some }i\neq j\}.</math><ref>{{cite journal |arxiv = 0806.4111|last1 = Farber|first1 = Michael|title = Topological complexity of configuration spaces|last2 = Grant|first2 = Mark|year = 2009|journal=[[Proceedings of the American Mathematical Society]]|volume=137|issue=5|pages=1841–1847|mr=2470845|doi=10.1090/S0002-9939-08-09808-0|s2cid = 16188638}}</ref>

This space is generally endowed with the subspace topology from the inclusion of <math>\operatorname{Conf}_n(X)</math> into <math>X^n</math>. It is also sometimes denoted <math>F(X, n)</math>, <math>F^n(X)</math>, or <math>\mathcal{C}^n(X)</math>.<ref name=":0" />

There is a natural [[Group action (mathematics)|action]] of the [[symmetric group]] <math>S_n</math> on the points in <math>\operatorname{Conf}_n(X)</math> given by

: <math>\begin{align}
  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{align}</math>

This action gives rise to the '''{{var|n}}<sup>th</sup> unordered configuration space of {{var|X}}''',

: <math>\operatorname{UConf}_n(X) := \operatorname{Conf}_n(X)/S_n,</math>

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 <math>\mathcal{UC}^n(X)</math>,<ref name=":0" /> <math>B_n(X)</math>, or <math>C_n(X)</math>. The collection of unordered configuration spaces over all <math>n</math> is the [[Ran space]], and comes with a natural topology.

=== Alternative formulations ===
For a topological space <math>X</math> and a finite set <math>S</math>, the '''configuration space of {{var|X}} with particles labeled by {{var|S}}''' is

: <math>\operatorname{Conf}_S(X) := \{f\mid f\colon S\hookrightarrow X\text{ is injective}\}.</math>

For <math>n\in\N</math>, define <math>\mathbf{n}:=\{1,2,\ldots,n\}</math>. Then the '''{{var|n}}<sup>th</sup> configuration space of ''X''''' is denoted simply <math>\operatorname{Conf}_n(X)</math>.<ref>{{Cite journal |arxiv=1612.08290|last1=Chettih|first1=Safia|title=The Homology of Configuration Spaces of Trees with Loops|journal=[[Algebraic & Geometric Topology]] |volume=18|issue=4|pages=2443–2469|last2=Lütgehetmann|first2=Daniel|year=2018|doi=10.2140/agt.2018.18.2443|s2cid=119168700}}</ref>

== Examples ==
* The space of ordered configuration of two points in <math>\mathbf{R}^2</math> is [[Homeomorphism|homeomorphic]] to the product of the Euclidean 3-space with a circle, i.e. <math>\operatorname{Conf}_2(\mathbf{R}^2)\cong \mathbf{R}^3\times S^1</math>.<ref name=":0" />
*More generally, the configuration space of two points in <math>\mathbf{R}^n</math> is [[Homotopy|homotopy equivalent]] to the sphere <math>S^{n-1}</math>.<ref>{{cite arXiv|last=Sinha|first=Dev|date=2010-02-20|title=The homology of the little disks operad |page=2 |eprint=math/0610236}}</ref>
*The configuration space of <math>n</math> points in <math>\mathbf{R}^2</math> is the classifying space of the <math>n</math>th [[braid group]] (see [[#Connection to braid groups|below]]).

== Connection to braid groups ==
{{Main|Braid group}}

The '''{{var|n}}-strand braid group''' on a [[Connected space|connected]] topological space {{var|X}} is 
:<math>B_n(X):=\pi_1(\operatorname{UConf}_n(X)),</math>
the [[fundamental group]] of the {{var|n}}<sup>th</sup> unordered configuration space of {{var|X}}. The '''{{var|n}}-strand pure braid group''' on {{var|X}} is<ref name=":0">{{Cite book|title=Braids|last=Ghrist|first=Robert|date=2009-12-01|publisher=World Scientific|isbn=9789814291408|editor-last=Berrick|editor-first=A. Jon|series=Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore|volume=19|pages=263–304|chapter=Configuration Spaces, Braids, and Robotics|doi=10.1142/9789814291415_0004|editor-last2=Cohen|editor-first2=Frederick R.|editor-last3=Hanbury|editor-first3=Elizabeth|editor-last4=Wong|editor-first4=Yan-Loi|editor-last5=Wu|editor-first5=Jie}}</ref>
:<math>P_n(X):=\pi_1(\operatorname{Conf}_n(X)).</math>

The first studied braid groups were the '''Artin braid groups''' <math>B_n\cong\pi_1(\operatorname{UConf}_n(\mathbf{R}^2))</math>. 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).<ref>{{cite book |series=[[Lecture Notes in Mathematics]] |volume=372 |first=Wilhelm |last=Magnus | author-link=Wilhelm Magnus |chapter=Braid groups: A survey |chapter-url=https://doi.org/10.1007%2FBFb0065203 |title=Proceedings of the Second International Conference on the Theory of Groups |publisher=Springer |year=1974 |isbn=978-3-540-06845-7 |pages=465|doi=10.1007/BFb0065203 }}</ref>

It follows from this definition and the fact that <math>\operatorname{Conf}_n(\mathbf{R}^2)</math> and <math>\operatorname{UConf}_n(\mathbf{R}^2)</math> are [[Eilenberg–MacLane space]]s of type <math>K(\pi,1)</math>, that the unordered configuration space of the plane <math>\operatorname{UConf}_n(\mathbf{R}^2)</math> is a  [[classifying space]] for the Artin braid group, and <math>\operatorname{Conf}_n(\mathbf{R}^2)</math> is a classifying space for the pure Artin braid group, when both are considered as [[discrete group]]s.<ref>{{Cite book|last=Arnold|first=Vladimir|author-link=Vladimir Arnold|others=Translated by [[Victor Vassiliev]]|title=Vladimir I. Arnold &mdash; Collected Works |chapter=The cohomology ring of the colored braid group |language=ru|volume=5|pages=227–231|doi=10.1007/978-3-642-31031-7_18|issn=0025-567X|mr=0242196|year=1969|isbn=978-3-642-31030-0|s2cid=122699084 }}</ref>

== Configuration spaces of manifolds ==
If the original space <math>X</math> is a [[manifold]], its ordered configuration spaces are open subspaces of the powers of <math>X</math> and are thus themselves manifolds. The configuration space of distinct unordered points is also a manifold, while the configuration space of ''not necessarily distinct''{{Clarify|reason=The "configuration space" of non-necessarily distinct points is not a configuration space, it's just the iterated cartesian product of the space with itself.|date=February 2019}} 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 <math> \pi\colon E_n\to C_n </math> which is a sub-bundle of the trivial bundle <math> C_n\times X\to C_n</math>, and which has the property that the fiber over each point <math> p\in C_n</math> is the ''n'' element subset of <math> X </math> classified by&nbsp;''p''.

=== Homotopy invariance ===
The homotopy type of configuration spaces is not [[homotopy invariant]]. For example, the spaces <math>\operatorname{Conf}_n(\mathbb R^m)</math> are not homotopy equivalent for any two distinct values of <math>m</math>: <math>\mathrm{Conf}_n(\mathbb{R}^0)</math> is empty for <math>n \ge 2</math>, <math>\operatorname{Conf}_n(\mathbb R)</math> is not connected for <math>n \ge 2</math>, <math>\operatorname{Conf}_n(\mathbb R^2)</math> is an [[Eilenberg–MacLane space]] of type <math>K(\pi,1)</math>, and <math>\operatorname{Conf}_n(\mathbb R^m)</math> is [[Simply connected space|simply connected]] for <math> m \geq 3</math>.

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 space]]s, and the configuration spaces of at least two points in them. That these configuration spaces are not homotopy equivalent was detected by [[Massey product]]s in their respective universal covers.<ref>{{citation|title=Configuration spaces are not homotopy invariant|year=2005|last1=Salvatore|last2=Longoni|first1=Paolo|first2=Riccardo|journal=Topology|volume=44|issue=2|pages=375&ndash;380|doi=10.1016/j.top.2004.11.002|arxiv=math/0401075|s2cid=15874513}}</ref> Homotopy invariance for configuration spaces of [[Simply connected space|simply connected]] closed manifolds remains open in general, and has been proved to hold over the base field <math>\mathbf{R}</math>.<ref>{{cite journal|last1=Campos|first1=Ricardo|last2=Willwacher|first2=Thomas|author-link2=Thomas Willwacher|date=2023|title=A model for configuration spaces of points|journal=Algebraic & Geometric Topology |volume=23 |issue=5 |pages=2029–2106 |doi=10.2140/agt.2023.23.2029 |arxiv=1604.02043 }}</ref><ref>{{cite journal|last=Idrissi|first=Najib|date=2016-08-29|title=The Lambrechts–Stanley Model of Configuration Spaces|url=https://archive.org/details/arxiv-1608.08054|journal=Inventiones Mathematicae|volume=216|pages=1–68|arxiv=1608.08054 |doi=10.1007/s00222-018-0842-9|bibcode=2016arXiv160808054I|s2cid=102354039}}</ref> Real homotopy invariance of simply connected compact [[Manifold with boundary|manifolds with simply connected boundary]] of dimension at least 4 was also proved.<ref>{{cite arXiv|last1=Campos|first1=Ricardo|last2=Idrissi|first2=Najib|last3=Lambrechts|first3=Pascal|last4=Willwacher|first4=Thomas|author-link4=Thomas Willwacher|date=2018-02-02|title=Configuration Spaces of Manifolds with Boundary|eprint=1802.00716|class=math.AT}}</ref>

== Configuration spaces of graphs ==
Some results are particular to configuration spaces of [[Graph (topology)|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.<ref name=":1">{{citation |last=Ghrist|first=Robert|author-link=Robert Ghrist|contribution=Configuration spaces and braid groups on graphs in robotics|title= Knots, braids, and mapping class groups—papers dedicated to Joan S. Birman|pages=29–40|series= AMS/IP Stud. Adv. Math.|volume=24|publisher=[[American Mathematical Society]]|location= Providence, RI| year=2001| arxiv=math/9905023|mr=1873106|title-link=Joan Birman}}</ref>

For any graph <math>\Gamma</math>, <math>\operatorname{Conf}_n(\Gamma)</math> is an Eilenberg–MacLane space of type <math>K(\pi,1)</math><ref name=":1" /> and [[Deformation retract|strong deformation retracts]] to a [[CW complex]] of dimension <math>b(\Gamma)</math>, where <math>b(\Gamma)</math> is the number of vertices of [[Degree (graph theory)|degree]] at least 3.<ref name=":1" /><ref>{{Cite journal|last1=Farley|first1=Daniel|last2=Sabalka|first2=Lucas|year=2005|title=Discrete Morse theory and graph braid groups|journal=[[Algebraic & Geometric Topology]]|volume=5|issue=3|pages=1075–1109|doi=10.2140/agt.2005.5.1075|arxiv=math/0410539|mr=2171804|s2cid=119715655}}</ref> Moreover, <math>\operatorname{UConf}_n(\Gamma)</math> and <math>\operatorname{Conf}_n(\Gamma)</math> deformation retract to [[Non-positive curvature|non-positively curved]] [[cubical complex]]es of dimension at most <math>\min(n, b(\Gamma))</math>.<ref>{{Cite journal|last=Świątkowski|first=Jacek|year=2001|title=Estimates for homological dimension of configuration spaces of graphs|journal=Colloquium Mathematicum|language=pl|volume=89|issue=1|pages=69–79|doi=10.4064/cm89-1-5|mr=1853416 |doi-access=free}}</ref><ref>{{cite thesis|first=Daniel|last= Lütgehetmann| title=Configuration spaces of graphs| degree=Master’s| publisher=[[Free University of Berlin]]| location=Berlin|year= 2014}}</ref>

== Configuration spaces of mechanical linkages ==
One also defines the configuration space of a mechanical linkage with the graph <math>\Gamma</math> 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 <math>n</math> rigid rods connected with revolute joints, the configuration space is the n-torus <math>T^n</math>.<ref>{{Cite journal|last1=Shvalb|first1=Nir|last2=Shoham|first2=Moshe|last3=Blanc|first3=David|year=2005|title=The configuration space of arachnoid mechanisms|journal=Forum Mathematicum|language=en|volume=17|issue=6|pages=1033–1042|doi=10.1515/form.2005.17.6.1033|s2cid=121995780}}</ref><ref>{{Cite book|last=Farber|first=Michael|year=2007|title=Invitation to Topological Robotics|publisher=american Mathematical Society}}</ref>
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).<ref>{{Cite journal|last1=Shvalb|first1=Nir|last2=Blanc|first2=David|year=2012|title=Generic singular configurations of linkages|journal=Topology and Its Applications|language=en|volume=159|issue=3|pages=877–890|doi=10.1016/j.topol.2011.12.003|doi-access=free|arxiv=1112.2334}}</ref>

== Compactification ==
The configuration space <math>\operatorname{Conf}_n(X)</math> of distinct points is non-compact, having ends where the points tend to approach each other (become confluent). Many geometric applications require compact spaces, so one would like to [[Compactification (mathematics)|compactify]] <math>\operatorname{Conf}_n(X)</math>, 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]],<ref>{{Cite journal |last1=Bott |first1=Raoul |author1-link=Raoul Bott|last2=Taubes |first2=Clifford |author2-link=Clifford Taubes|date=1994-10-01 |title=On the self-linking of knots |url=http://dx.doi.org/10.1063/1.530750 |journal=[[Journal of Mathematical Physics]] |volume=35 |issue=10 |pages=5247–5287 |doi=10.1063/1.530750 |issn=0022-2488}}</ref> as well as [[William Fulton (mathematician)|William Fulton]] and [[Robert MacPherson (mathematician)|Robert MacPherson]].<ref>{{Cite journal |last1=Fulton |first1=William |author1-link=William Fulton (mathematician)|last2=MacPherson |first2=Robert |author2-link=Robert MacPherson (mathematician)|date=January 1994 |title=A Compactification of Configuration Spaces |url=http://dx.doi.org/10.2307/2946631 |journal=[[Annals of Mathematics]] |volume=139 |issue=1 |pages=183 |doi=10.2307/2946631 |jstor=2946631 |issn=0003-486X}}</ref>

==See also==
{{Portal|Mathematics}}
*[[Configuration space (physics)]]
*[[State space (physics)]]

== References ==
{{Reflist|30em}}

{{Topology}}

{{DEFAULTSORT:Configuration Space}}
[[Category:Manifolds]]
[[Category:Topology]]
[[Category:Algebraic topology]]