configuration space (mathematics)

For a topological space $X$ and a positive integer $n$, let $X^n$ be the Cartesian product of $n$ copies of $X$, equipped with the product topology. The n^th (ordered) configuration space of $X$ is the set of n-tuples of pairwise distinct points in $X$:

:$\backslash operatorname\{Conf\}\_n(X)\; :=\; X^n\; \backslash smallsetminus\; \backslash \{(x\_1,x\_2,\backslash ldots,x\_n)\backslash in\; X^n\; \backslash mid\; x\_i=\; x\_j\backslash \; \backslash text\{for\; some\; \}i\backslash neq\; j\backslash \}.${{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}}

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

There is a natural action of the symmetric group $S\_n$ on the points in $\backslash operatorname\{Conf\}\_n(X)$ given by

: $\backslash begin\{align\}$

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}

This action gives rise to the {{var|n}}^th unordered configuration space of {{var|X}},

: $\backslash operatorname\{UConf\}\_n(X)\; :=\; \backslash operatorname\{Conf\}\_n(X)/S\_n,$

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

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

: $\backslash operatorname\{Conf\}\_S(X)\; :=\; \backslash \{f\backslash mid\; f\backslash colon\; S\backslash hookrightarrow\; X\backslash text\{\; is\; injective\}\backslash \}.$

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

• The space of ordered configuration of two points in $\backslash mathbf\{R\}^2$ is homeomorphic to the product of the Euclidean 3-space with a circle, i.e. $\backslash operatorname\{Conf\}\_2(\backslash mathbf\{R\}^2)\backslash cong\; \backslash mathbf\{R\}^3\backslash times\; S^1$.
• More generally, the configuration space of two points in $\backslash mathbf\{R\}^n$ is homotopy equivalent to the sphere $S^\{n-1\}$.{{cite arXiv|last=Sinha|first=Dev|date=2010-02-20|title=The homology of the little disks operad |page=2 |eprint=math/0610236}}
• The configuration space of $n$ points in $\backslash mathbf\{R\}^2$ is the classifying space of the $n$th braid group (see below).

{{Main|Braid group}}

The {{var|n}}-strand braid group on a connected topological space {{var|X}} is

:$B\_n(X):=\backslash pi\_1(\backslash operatorname\{UConf\}\_n(X)),$

the fundamental group of the {{var|n}}^th unordered configuration space of {{var|X}}. The {{var|n}}-strand pure braid group on {{var|X}} is{{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}}

:$P\_n(X):=\backslash pi\_1(\backslash operatorname\{Conf\}\_n(X)).$

The first studied braid groups were the Artin braid groups $B\_n\backslash cong\backslash pi\_1(\backslash operatorname\{UConf\}\_n(\backslash mathbf\{R\}^2))$. 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).{{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 }}

It follows from this definition and the fact that $\backslash operatorname\{Conf\}\_n(\backslash mathbf\{R\}^2)$ and $\backslash operatorname\{UConf\}\_n(\backslash mathbf\{R\}^2)$ are Eilenberg–MacLane spaces of type $K(\backslash pi,1)$, that the unordered configuration space of the plane $\backslash operatorname\{UConf\}\_n(\backslash mathbf\{R\}^2)$ is a classifying space for the Artin braid group, and $\backslash operatorname\{Conf\}\_n(\backslash mathbf\{R\}^2)$ is a classifying space for the pure Artin braid group, when both are considered as discrete groups.{{Cite book|last=Arnold|first=Vladimir|author-link=Vladimir Arnold|others=Translated by Victor Vassiliev|title=Vladimir I. Arnold — 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 }}

If the original space $X$ is a manifold, its ordered configuration spaces are open subspaces of the powers of $X$ and are thus themselves manifolds. The configuration space of distinct unordered points is also a manifold, while the configuration space of not necessarily distinct{{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 $\backslash pi\backslash colon\; E\_n\backslash to\; C\_n$ which is a sub-bundle of the trivial bundle $C\_n\backslash times\; X\backslash to\; C\_n$, and which has the property that the fiber over each point $p\backslash in\; C\_n$ is the n element subset of $X$ classified by p.

The homotopy type of configuration spaces is not homotopy invariant. For example, the spaces $\backslash operatorname\{Conf\}\_n(\backslash mathbb\; R^m)$ are not homotopy equivalent for any two distinct values of $m$: $\backslash mathrm\{Conf\}\_n(\backslash mathbb\{R\}^0)$ is empty for $n\; \backslash ge\; 2$, $\backslash operatorname\{Conf\}\_n(\backslash mathbb\; R)$ is not connected for $n\; \backslash ge\; 2$, $\backslash operatorname\{Conf\}\_n(\backslash mathbb\; R^2)$ is an Eilenberg–MacLane space of type $K(\backslash pi,1)$, and $\backslash operatorname\{Conf\}\_n(\backslash mathbb\; R^m)$ is simply connected for $m\; \backslash geq\; 3$.

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 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.{{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=375–380|doi=10.1016/j.top.2004.11.002|arxiv=math/0401075|s2cid=15874513}} Homotopy invariance for configuration spaces of simply connected closed manifolds remains open in general, and has been proved to hold over the base field $\backslash mathbf\{R\}$.{{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 }}{{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}} Real homotopy invariance of simply connected compact manifolds with simply connected boundary of dimension at least 4 was also proved.{{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}}

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.{{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}}

For any graph $\backslash Gamma$, $\backslash operatorname\{Conf\}\_n(\backslash Gamma)$ is an Eilenberg–MacLane space of type $K(\backslash pi,1)$ and strong deformation retracts to a CW complex of dimension $b(\backslash Gamma)$, where $b(\backslash Gamma)$ is the number of vertices of degree at least 3.{{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}} Moreover, $\backslash operatorname\{UConf\}\_n(\backslash Gamma)$ and $\backslash operatorname\{Conf\}\_n(\backslash Gamma)$ deformation retract to non-positively curved cubical complexes of dimension at most $\backslash min(n,\; b(\backslash Gamma))$.{{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}}{{cite thesis|first=Daniel|last= Lütgehetmann| title=Configuration spaces of graphs| degree=Master’s| publisher=Free University of Berlin| location=Berlin|year= 2014}}

One also defines the configuration space of a mechanical linkage with the graph $\backslash Gamma$ 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 $n$ rigid rods connected with revolute joints, the configuration space is the n-torus $T^n$.{{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}}{{Cite book|last=Farber|first=Michael|year=2007|title=Invitation to Topological Robotics|publisher=american Mathematical Society}}

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).{{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}}

The configuration space $\backslash operatorname\{Conf\}\_n(X)$ 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 compactify $\backslash operatorname\{Conf\}\_n(X)$, 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,{{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 |bibcode=1994JMP....35.5247B |issn=0022-2488|url-access=subscription }} as well as William Fulton and Robert MacPherson.{{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|url-access=subscription }}

{{Portal|Mathematics}}

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

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