Contents

# Contents

## Idea

In mathematics typically by default the term “configuration space” of a topological space $X$ refers to the topological space of pairwise distinct points in $X$, also called Fadell's configuration space, for emphasis.

In principle many other kinds of configurations and the spaces these form may be referred to by “configuration space”, notably in physics the usage is in a broader sense, see at configuration space (physics).

## Definition

###### Definition

(configuration spaces of points)

Let $X$ be a manifold, possibly with boundary. For $n \in \mathbb{N}$, the configuration space of $n$ points in $X$ disappearing at the boundary is the topological space

$\mathrm{Conf}_{n}(X) \;\coloneqq\; \Big( \big( X^n \setminus \mathbf{\Delta}_X^n \big) / \partial(X^n) \Big) /\Sigma(n) \,,$

where $\mathbf{\Delta}_X^n : = \{(x^i) \in X^n | \underset{i,j}{\exists} (x^i = x^j) \}$ is the fat diagonal in $X^n$ and where $\Sigma(n)$ denotes the evident action of the symmetric group by permutation of factors of $X$ inside $X^n$.

More generally, let $Y$ be another manifold, possibly with boundary. For $n \in \mathbb{N}$, the configuration space of $n$ points in $X \times Y$ vanishing at the boundary and distinct as points in $X$ is the topological space

$\mathrm{Conf}_{n}(X,Y) \;\coloneqq\; \Big( \big( ( X^n \setminus \mathbf{\Delta}_X^n ) \times Y^n \big) /\Sigma(n) \Big) / \partial(X^n \times Y^n)$

where now $\Sigma(n)$ denotes the evident action of the symmetric group by permutation of factors of $X \times Y$ inside $X^n \times Y^n \simeq (X \times Y)^n$.

This more general definition reduces to the previous case for $Y = \ast \coloneqq \mathbb{R}^0$ being the point:

$\mathrm{Conf}_n(X) \;=\; \mathrm{Conf}_n(X,\ast) \,.$

Finally the configuration space of an arbitrary number of points in $X \times Y$ vanishing at the boundary and distinct already as points of $X$ is the quotient topological space of the disjoint union space

$Conf\left( X, Y\right) \;\coloneqq\; \left( \underset{n \in \mathbb{n}}{\sqcup} \big( ( X^n \setminus \mathbf{\Delta}_X^n ) \times Y^k \big) /\Sigma(n) \right)/\sim$

by the equivalence relation $\sim$ given by

$\big( (x_1, y_1), \cdots, (x_{n-1}, y_{n-1}), (x_n, y_n) \big) \;\sim\; \big( (x_1, y_1), \cdots, (x_{n-1}, y_{n-1}) \big) \;\;\;\; \Leftrightarrow \;\;\;\; (x_n, y_n) \in \partial (X \times Y) \,.$

This is naturally a filtered topological space with filter stages

$Conf_{\leq n}\left( X, Y\right) \;\coloneqq\; \left( \underset{k \in \{1, \cdots, n\}}{\sqcup} \big( ( X^k \setminus \mathbf{\Delta}_X^k ) \times Y^k \big) /\Sigma(k) \right)/\sim \,.$

The corresponding quotient topological spaces of the filter stages reproduces the above configuration spaces of a fixed number of points:

$Conf_n(X,Y) \;\simeq\; Conf_{\leq n}(X,Y) / Conf_{\leq (n-1)}(X,Y) \,.$
###### Remark

(comparison to notation in the literature)

The above Def. is less general but possibly more suggestive than what is considered for instance in Bödigheimer 87. Concretely, we have the following translations of notation:

$\array{ \text{ here: } && \array{ \text{ Segal 73,} \\ \text{ Snaith 74}: } && \text{ Bödigheimer 87: } \\ \\ Conf(\mathbb{R}^d,Y) &=& C_d( Y/\partial Y ) &=& C( \mathbb{R}^d, \emptyset; Y ) \\ \mathrm{Conf}_n\left( \mathbb{R}^d \right) & = & F_n C_d( S^0 ) / F_{n-1} C_d( S^0 ) & = & D_n\left( \mathbb{R}^d, \emptyset; S^0 \right) \\ \mathrm{Conf}_n\left( \mathbb{R}^d, Y \right) & = & F_n C_d( Y/\partial Y ) / F_{n-1} C_d( Y/\partial Y ) & = & D_n\left( \mathbb{R}^d, \emptyset; Y/\partial Y \right) \\ \mathrm{Conf}_n( X ) && &=& D_n\left( X, \partial X; S^0 \right) \\ \mathrm{Conf}_n( X, Y ) && &=& D_n\left( X, \partial X; Y/\partial Y \right) }$

Notice here that when $Y$ happens to have empty boundary, $\partial Y = \emptyset$, then the pushout

$X / \partial Y \coloneqq Y \underset{\partial Y}{\sqcup} \ast$

is $Y$ with a disjoint basepoint attached. Notably for $Y =\ast$ the point space, we have that

$\ast/\partial \ast = S^0$

is the 0-sphere.

A slight variation of the definition is sometimes useful:

###### Definition

(configuration space of $dim(X)$-disks)

In the situation of Def. , with $X$ a manifold of dimension $dim(X) \in \mathbb{N}$

$DiskConf(X,A) \longrightarrow Conf(X,A)$

be, on the left, the labeled configuration space of joint embeddings of tuples

$\left( D^{dim(X)} \overset{ \iota_i }{\hookrightarrow} X \right)$

of $dim(X)$-dimensional disks/closed balls into $X$, with identifications as in Def. (in particular the disks centered at the basepoint are quotiented out) and with the comparison map sending each disk to its center.

This map is evidently a deformation retraction hence in particular a homotopy equivalence.

## Properties

### Iterated loop spaces of iterated suspensions

###### Proposition

For

1. $d \in \mathbb{N}$, $d \geq 1$ a natural number with $\mathbb{R}^d$ denoting the Cartesian space/Euclidean space of that dimension,

2. $Y$ a manifold, with non-empty boundary so that $Y / \partial Y$ is connected,

the electric field map/scanning map constitutes a homotopy equivalence

$Conf\left( \mathbb{R}^d, Y \right) \overset{scan}{\longrightarrow} \Omega^d \Sigma^d (Y/\partial Y)$

between

1. the configuration space of arbitrary points in $\mathbb{R}^d \times Y$ vanishing at the boundary (Def. )

2. the $d$-fold loop space of the $d$-fold reduced suspension of the quotient space $Y / \partial Y$ (regarded as a pointed topological space with basepoint $[\partial Y]$).

In particular when $Y = \mathbb{D}^k$ is the closed ball of dimension $k \geq 1$ this gives a homotopy equivalence

$Conf\left( \mathbb{R}^d, \mathbb{D}^k \right) \overset{scan}{\longrightarrow} \Omega^d S^{ d + k }$

with the $d$-fold loop space of the (d+k)-sphere.

###### Proposition

(stable splitting of mapping spaces out of Euclidean space/n-spheres)

For

1. $d \in \mathbb{N}$, $d \geq 1$ a natural number with $\mathbb{R}^d$ denoting the Cartesian space/Euclidean space of that dimension,

2. $Y$ a manifold, with non-empty boundary so that $Y / \partial Y$ is connected,

there is a stable weak homotopy equivalence

$\Sigma^\infty Conf(\mathbb{R}^d, Y) \overset{\simeq}{\longrightarrow} \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d, Y)$

between

1. the suspension spectrum of the configuration space of an arbitrary number of points in $\mathbb{R}^d \times Y$ vanishing at the boundary and distinct already as points of $\mathbb{R}^d$ (Def. )

2. the direct sum (hence: wedge sum) of suspension spectra of the configuration spaces of a fixed number of points in $\mathbb{R}^d \times Y$, vanishing at the boundary and distinct already as points in $\mathbb{R}^d$ (also Def. ).

Combined with the stabilization of the electric field map/scanning map homotopy equivalence from Prop. this yields a stable weak homotopy equivalence

$Maps_{cp}(\mathbb{R}^d, \Sigma^d (Y / \partial Y)) = Maps^{\ast/}( S^d, \Sigma^d (Y / \partial Y)) = \Omega^d \Sigma^d (Y/\partial Y) \underoverset{\Sigma^\infty scan}{\simeq}{\longrightarrow} \Sigma^\infty Conf(\mathbb{R}^d, Y) \overset{\simeq}{\longrightarrow} \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d, Y)$

between the latter direct sum and the suspension spectrum of the mapping space of pointed continuous functions from the d-sphere to the $d$-fold reduced suspension of $Y / \partial Y$.

In fact by Bödigheimer 87, Example 5 this equivalence still holds with $Y$ treated on the same footing as $\mathbb{R}^d$, hence with $Conf_n(\mathbb{R}^d, Y)$ on the right replaced by $Conf_n(\mathbb{R}^d \times Y)$ in the well-adjusted notation of Def. :

$Maps_{cp}(\mathbb{R}^d, \Sigma^d (Y / \partial Y)) = Maps^{\ast/}( S^d, \Sigma^d (Y / \partial Y)) \overset{\simeq}{\longrightarrow} \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d \times Y)$

### Classifying space of the symmetric group

Let $X= \mathbb{R}^\infty$. Then

• the unordered configuration space of $n$ points in $\mathbb{R}^\infty$ is a model for the classifying space $B \Sigma(n)$ of the symmetric group $\Sigma(n)$;

• the ordered configuration space of $n$ points, equipped with the canonical $\Sigma(n)$-action, is a model for the $\Sigma(n)$-universal principal bundle.

$\,$

### James construction

The James construction of $X$ is homotopy equivalent to the configuration space $C(\mathbb{R}^1, X)$ of points in the real line with “charges” taking values in $X$.

$\,$

### Action by little $n$-disk operad and by Goodwillie derivatives

Under some conditions and with suitable degrees/shifts, configuration spaces of points canonically have the structure of algebras over an operad over the little n-disk operad and the Goodwillie derivatives of the identitity functor?.

For more see there

## Occurrences and Applications

### Compactification

The Fulton-MacPherson compactification of configuration spaces of points in $\mathbb{R}^d$ serves to exhibit them as models for the little n-disk operad.

### Stable splitting of mapping spaces

The stable splitting of mapping spaces says that suspension spectra of suitable mapping spaces are equivalently wedge sums of suspension spectra of configuration spaces of points.

### Correlators as differential forms on configiration spaces

In Euclidean field theory the correlators are often regarded as distributions of several variables with singularities on the fat diagonal. Hence they become non-singular distributions after restriction of distributions to the corresponding configuration space of points.

For more on this see at correlators as differential forms on configuration spaces of points.

## References

### General

General survey includes

• Edward Fadell, Lee Neuwirth, Configuration spaces Math. Scand. 10 (1962) 111-118, MR141126, pdf

• Edward R. Fadell, Sufian Y. Husseini, Geometry and topology of configuration spaces, Springer Monographs in Mathematics (2001), MR2002k:55038, xvi+313 pp.

• F. R. Cohen, S. Gitler, On loop spaces of configuration spaces, Trans. Amer. Math. Soc. 354 (2002), no. 5, 1705–1748, MR2002m:55020

• Sadok Kallel, Ines Saihi, Homotopy Groups of Diagonal Complements, Algebr. Geom. Topol. 16 (2016) 2949-2980 (arXiv:1306.6272)

### Electric field map/Scanning map and cohomotopy

The electric field map/scanning map and hence the relation of configuration spaces to cohomotopy goes back to

• Peter May, The geometry of iterated loop spaces, Springer 1972 (pdf)

• Graeme Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973), 213–221. MR 0331377 (pdf)

Generalization of these constructions and results is due to

and generalization to equivariant cohomology is discussed in

• Colin Rourke, Brian Sanderson, Equivariant Configuration Spaces, 62(2), October 2000, pp. 544-552 (pdf)

For relation to instantons via topological Yang-Mills theory:

In speculation regarding Galois theory over the sphere spectrum:

### Algebra structure over little $dim(X)$-disk operad

• Martin Markl, A compactification of the real configuration space as an operadic completion, J. Algebra 215 (1999), no. 1, 185–204

### Stable splitting of mapping spaces

The appearance of configuration spaces as summands in stable splittings of mapping spaces is originally due to

• Victor Snaith, A stable decomposition of $\Omega^n S^n X$, Journal of the London Mathematical Society 7 (1974), 577 - 583 (pdf)

An alternative proof is due to

Review and generalization is in

and the relation to the Goodwillie-Taylor tower of mapping spaces is pointed out in

### In Goodwillie-calculus

The configuration spaces of a space $X$ appear as the Goodwillie derivatives of its mapping space/nonabelian cohomology-functor $Maps(X,-)$:

• Greg Arone, A generalization of Snaith-type filtration, Transactions of the American Mathematical Society 351.3 (1999): 1123-1150. (pdf)

• Michael Ching, Calculus of Functors and Configuration Spaces, Conference on Pure and Applied Topology Isle of Skye, Scotland, 21-25 June, 2005 (pdf)

### Compactification

A compactification of configuration spaces of points was introduced in

Review includes

This underlies the models of configuration spaces by graph complexes, see there for more.

### Cohomology of configuration spaces

{ReferencesCohomology}

General discussion of ordinary homology/ordinary cohomology of configuration spaces of points includes

• Carl-Friedrich Bödigheimer, F. Cohen, L. Taylor, On the homology of configuration spaces, Topology Vol. 28 No. 1, p. 111-123 1989 (pdf)

• Ben Knudsen, Betti numbers and stability for configuration spaces via factorization homology, Algebr. Geom. Topol. 17 (2017) 3137-3187 (arXiv:1405.6696)

• Christoph Schiessl, Cohomology of Configuration Spaces (pdf)

### Cohomology modeled by graph complexes

That the de Rham cohomology of (the Fulton-MacPherson compactification of) configuration spaces of points may be modeled by graph complexes (exhibiting formality of the little n-disk operad) is due to

niecely reviewed in

Further discussion of the graph complex as a model for the de Rham cohomology of configuration spaces of points is in

Last revised on February 7, 2019 at 08:48:45. See the history of this page for a list of all contributions to it.